❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❯♠ ❡st✉❞♦ s♦❜r❡
p
✲❣r✉♣♦s ✜♥✐t♦s
♣♦✇❡r❢✉❧ ❡ ♣♦t❡♥t
◆❛t❤á❧✐❛ ◆♦❣✉❡✐r❛ ●♦♥ç❛❧✈❡s
❇r❛sí❧✐❛
◆❛t❤á❧✐❛ ◆♦❣✉❡✐r❛ ●♦♥ç❛❧✈❡s
❯♠ ❡st✉❞♦ s♦❜r❡
p
✲❣r✉♣♦s ✜♥✐t♦s
♣♦✇❡r❢✉❧ ❡ ♣♦t❡♥t
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❊❙❚❘❊ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿
Pr♦❢✳ ❉r✳ ❊♠❡rs♦♥ ❋❡rr❡✐r❛ ❞❡ ▼❡❧♦
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
GG643e
Gonçalves, Nathália Nogueira
Um estudo sobre p-grupos finitos powerful e potent / Nathália Nogueira Gonçalves; orientador Emerson Ferreira de Melo. -- Brasília, 2017. 105 p.
Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2017.
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛ ✈✐tór✐❛ ❛❧❝❛♥ç❛❞❛✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ❡ às ♠✐♥❤❛s ✐r♠ãs ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛ ♥❡ss❛ ❝❛♠✐♥❤❛❞❛✳ ◆❡♥❤✉♠❛ ♣❛❧❛✈r❛ ❞❡s❝r❡✈❡ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❡ ♠❡✉ ❛♠♦r ♣♦r ✈♦❝ês✳
❆♦ ♠❡✉ ❛♠♦r✱ ❘❛❢❛❡❧✱ ♣♦r ❡st❛r s❡♠♣r❡ ❝♦♠✐❣♦✱ ♣♦r ❛❣✉❡♥t❛r ♠❡✉s ❝❤♦r♦s✱ ❛✢✐çõ❡s ❡ ♠❛✉ ❤✉♠♦r ❞✉r❛♥t❡ ❡ss❡ t❡♠♣♦ ❡♠ ❇r❛sí❧✐❛✳ ❊ ♣♦r ♠❛✐s ❡ss❡ ♣❛ss♦ ❥✉♥t♦s✳
➚ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦ ♠❡✉ ❆✈ô✱ ♠❡✉s t✐♦s ❙ér❣✐♦✱ ❏❛❝❦ ❡ ❘❡♥❛t♦✱ ♣❡❧❛s ♦r❛çõ❡s ❡ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ s❡♠♣r❡✳ ❆♦s ♠❡✉s ♣r✐♠♦s✱ ●❛❜r✐❡❧ ❡ ❇❡♥✐s❛✱ q✉❡ sã♦ ♠❡✉s ✐r♠ã♦s ❞❡ ❝♦r❛çã♦✳ ❆♦ ♠❡✉ ❛✜❧❤❛❞♦✱ ❘❛❢❛❡❧✱ ♣❡❧❛ ❛❧❡❣r✐❛✳
❆♦s ♠❡✉s q✉❡r✐❞♦s ❛♠✐❣♦s ❞❡ ❖✉r♦ Pr❡t♦ ❡ ❞♦ ❈♦❧é❣✐♦ ❙✐♥❛♣s❡ ♣♦r t♦❞♦ ♦ ❝❛r✐♥❤♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❙✉ ❡ ❚✐❛♥②✱ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❡ s❡♠♣r❡✳
❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❯❋❖P✱ q✉❡ t❛♥t♦ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡s✲ s♦r❡s ❙❡❜❛st✐ã♦ ▼❛rt✐♥s✱ ❏❛♠✐❧ ❋❡rr❡✐r❛ ❡ ●✉st❛✈♦ ❙♦✉③❛ ♣♦r t♦❞♦ ♦ ✐♥❝❡♥t✐✈♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❊♠❡rs♦♥ ❋❡rr❡✐r❛ ❞❡ ▼❡❧♦ ♣♦r t♦❞♦s ♦s ❡♥s✐♥❛♠❡♥t♦s✱ ♣❛❝✐ê♥❝✐❛✱ ❞✐s♣♦s✐çã♦ ❡ ❞❡❞✐❝❛çã♦✳ ❖❜r✐❣❛❞❛ t❛♠❜é♠ ♣❡❧❛s ót✐♠❛s ❝♦♥✈❡rs❛s ♣♦❧ít✐❝❛s ❡ ♣❡ss♦❛✐s✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ♣❛rt✐❝✐♣❛♥t❡s ❞❛ ❜❛♥❝❛ ❏❤♦♥❡ ❈❛❧❞❡✐r❛ ❙✐❧✈❛ ❡ ❘❛✐♠✉♥❞♦ ❞❡ ❆r❛ú❥♦ ❇❛st♦s ❏ú♥✐♦r ♣♦r ❛❝❡✐t❛r❡♠ ♦ ❝♦♥✈✐t❡ ❡ t❛♠❜é♠ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❇ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s tr❛♥s✲ ♠✐t✐❞♦s✳ ❊♠ ❡s♣❡❝✐❛❧ ♦s ♣r♦❢❡ss♦r❡s ❈r✐st✐♥❛ ❆❝❝✐❛rr✐✱ ❉❛♥✐❡❧❛ ❆♠❛t♦✱ ❊♠❡rs♦♥ ❞❡ ▼❡❧♦✱ ▼❛rt✐♥♦ ●❛r♦♥③✐ ❡ ◆♦r❛í ❘♦❝❝♦✱ ♣♦r ♠❡ ♠♦str❛r❡♠ ♦ q✉❛♥t♦ ❛ ➪❧❣❡❜r❛ é ❢❛s❝✐♥❛♥t❡✳
❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ♣♦r t♦❞♦ ♦ ❛❝♦❧❤✐♠❡♥t♦ ❡ s✐♠♣❛t✐❛✳ ➚ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦✱ ♣❡❧❛s ✈ár✐❛s ❝♦♥✈❡rs❛s ❡ r✐s❛❞❛s✳ ❖❜r✐❣❛❞❛ ❛ t♦❞♦s ♦s ♣r❡s❡♥t❡s ♥❛ ♠✐♥❤❛ ❛♣r❡s❡♥t❛çã♦✱ ♣❡❧❛ ❢♦rç❛ ❡ t♦r❝✐❞❛✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❆❧❡①❛♥❞r❡✱ ❆♥❛ P❛✉❧❛✱ ❆♥♥❛ ❈❛r♦❧✐♥❛✱ ❇r✉♥♦✱ ❈❤r✐st❡✱ ▲✉♠❡♥❛✱ ❘❡❣✐❛♥❡✱ ❙❛r❛ ❡ ❲❡❧✐♥t♦♥✳ P♦r ❡st❛r❡♠ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s✳
❆❣r❛❞❡ç♦ ❛♦ ❈◆Pq ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ à ❡st❡ tr❛❜❛❧❤♦✳
❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ❝❡rt❛ ❢♦r♠❛ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝❤❡❣❛r ❛té ❛q✉✐✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ p✲❣r✉♣♦s ✜♥✐t♦s✳ ❉❡♥tr❡ ❛s ♠✉✐t❛s ♣r♦♣r✐❡✲ ❞❛❞❡s q✉❡ ✈❡r❡♠♦s✱ ❞❡st❛❝❛♠♦s ♦ ❡st✉❞♦ s♦❜r❡ ❛ ❡str✉t✉r❛ ♣♦✇❡r ❛❜❡❧✐❛♥ ❞♦s s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ✉♠p✲❣r✉♣♦ ✜♥✐t♦ ♣♦t❡♥t✱ q✉❡ ❢♦✐ ❡st✉❞❛❞❛ ❛tr❛✈és ❞♦ ❛rt✐❣♦ ✧❖♥ t❤❡ str✉❝t✉r❡ ♦❢ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ♦❢ ♣♦t❡♥t ♣✲❣r♦✉♣s✧✳ ❊ t❛♠❜é♠ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ s❡r ♣♦✇❡r❢✉❧ ♦❜t✐❞❛ ♥♦ ❛rt✐❣♦ ✧❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣♦✇❡r❢✉❧ ♣✲❣r♦✉♣s✧✳
P❛❧❛✈r❛s✲❈❤❛✈❡s✿ p✲❣r✉♣♦s ✜♥✐t♦s❀ ♣♦✇❡r❢✉❧❀ ♣♦t❡♥t✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ st✉❞② ✜♥✐t❡ p✲❣r♦✉♣s✳ ❆♠♦♥❣ t❤❡ ♣r♦♣❡rt✐❡s✱ ✇❡ ❤✐❣❤❧✐❣❤t t❤❡ ♣♦✇❡r ❛❜❡❧✐❛♥ str✉❝t✉r❡ ♦❢ ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❛ ✜♥✐t❡ ♣♦t❡♥tp✲❣r♦✉♣✱ ✇❤✐❝❤ ✇❛s st✉❞✐❡❞ ✐♥ t❤❡ ♣❛♣❡r ✧❖♥ t❤❡ str✉❝t✉r❡ ♦❢ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ♦❢ ♣♦t❡♥t ♣✲❣r♦✉♣s✧✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❛ ✜♥✐t❡ p✲❣r♦✉♣ t♦ ❜❡ ❛ ♣♦✇❡r❢✉❧ p✲❣r♦✉♣ ♣r♦✈❡❞ ✐♥ t❤❡ ♣❛♣❡r ✧❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣♦✇❡r❢✉❧ ♣✲❣r♦✉♣s✧✳
❑❡②✲❲♦r❞s✿ ✜♥✐t❡ p✲❣r♦✉♣❀ ♣♦✇❡r❢✉❧❀ ♣♦t❡♥t✳
◆♦t❛çõ❡s
⌊r⌋ ❖ ♠❛✐♦r ✐♥t❡✐r♦ q✉❡ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ r✳ ⌈r⌉ ❖ ♠❡♥♦r ✐♥t❡✐r♦ q✉❡ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❞♦ q✉❡ r✳ o(x) ❖r❞❡♠ ❞♦ ❡❧❡♠❡♥t♦ x✳
xy y−1xy✳
[x, y] x−1y−1xy✳
[x1, . . . , xn] [[x1, . . . , xn−1], xn]✳
|G| ❖r❞❡♠ ❞♦ ❣r✉♣♦ G✳
d(G) ◆ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦G✳
H 6G ❍ ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ G✳
hXi ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ X✳
[H1, H2] h[x, y]|x∈H1, y ∈H2i✳
[H1, . . . , Hn] [[H, . . . , Hn−1], Hn]✳
[H1,kH2] [H1, H2, . . . , H2]✱ H2 ❛♣❛r❡❝❡ k ✈❡③❡s✳
NG(H) ◆♦r♠❛❧✐③❛❞♦r ❞♦ s✉❜❣r✉♣♦ H ♥♦ ❣r✉♣♦G✳
CG(H) ❈❡♥tr❛❧✐③❛❞♦r ❞♦ s✉❜❣r✉♣♦ H ♥♦ ❣r✉♣♦G✳
Z(G) ❈❡♥tr♦ ❞♦ ❣r✉♣♦ G✳
|G:H| ❮♥❞✐❝❡ ❞♦ s✉❜❣r✉♣♦ H ♥♦ ❣r✉♣♦ G✳
N EG N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞♦ ❣r✉♣♦ G✳
Φ(G) ❙✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ ❞♦ ❣r✉♣♦ G✳
γn(G) n✲és✐♠♦ t❡r♠♦ ❞❛ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r ❞♦ ❣r✉♣♦G✳
[G, G] =G′ ❙✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞♦ ❣r✉♣♦ G✳
Gn ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣❡❧❛s n✲és✐♠❛s ♣♦tê♥❝✐❛s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G✳
G{n} ❈♦♥❥✉♥t♦ ❞❛sn✲és✐♠❛s ♣♦tê♥❝✐❛s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G✳
Ωn(G) ❙✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ Gq✉❡ ♣♦ss✉❡♠ ♦r❞❡♠
♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ pn✳
Ω{n}(G) ❈♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ Gq✉❡ ♣♦ss✉❡♠ ♦r❞❡♠ ♠❡♥♦r ♦✉
✐❣✉❛❧ ❛ pn✳
Fp ❈♦r♣♦ ✜♥✐t♦ ❝♦♠ p ❡❧❡♠❡♥t♦s✳
Fp[t] ❆♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ♥❛ ✐♥❝ó❣♥✐t❛ t ❡ ❝♦❡✜❝✐❡♥t❡s ❡♠ Fp✳
❙✉♠ár✐♦
◆♦t❛çõ❡s ✶
■♥tr♦❞✉çã♦ ✹
✶ Pr❡❧✐♠✐♥❛r❡s ✽
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✶ ❈♦♠✉t❛❞♦r❡s ❡ s✉❜❣r✉♣♦s ❣❡r❛❞♦s ♣♦r ❝♦♠✉t❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✷ ●r✉♣♦s ♥✐❧♣♦t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸ ❖ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✹ ❋ór♠✉❧❛ ❞❡ ❈♦♠♣✐❧❛çã♦ ❞❡ ❍❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❆♥é✐s ❡ ▼ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❆❧❣✉♠❛s ❢❛♠í❧✐❛s ❞❡ p✲❣r✉♣♦s ✜♥✐t♦s ✶✻
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ p✲❣r✉♣♦s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ p✲❣r✉♣♦s ❞❡ ❝❧❛ss❡ ♠❛①✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✹ p✲❣r✉♣♦s ♣♦✇❡r❢✉❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✺ p✲❣r✉♣♦s ♣♦t❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✸ ❘❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s s♦❜r❡ p✲❣r✉♣♦s ♣♦t❡♥t ✺✻
✸✳✶ ❙✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ✉♠ p✲❣r✉♣♦ ♣♦t❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❊str✉t✉r❛ ❞❡ s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ✉♠ p✲❣r✉♣♦ ♣♦t❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✹ ❘❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s s♦❜r❡ p✲❣r✉♣♦s ♣♦✇❡r❢✉❧ ✽✶
✺ ❯♠❛ ❢❛♠í❧✐❛ ❞❡ ❡①❡♠♣❧♦s ✾✷ ✺✳✶ Pr❡❧✐♠✐♥❛r❡s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❢❛♠í❧✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✺✳✷ ❋❛♠í❧✐❛ ❞❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
■♥tr♦❞✉çã♦
❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ❝♦♠ p ✉♠ ♣r✐♠♦✳ ❯s❛♥❞♦ ♦ ❤♦♠♦♠♦r✜s♠♦ φ :G → G ❞❛❞♦ ♣♦r φ(g) = gpi
é ♣♦ssí✈❡❧ ✈❡r q✉❡ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧i ✈❛❧❡♠ ♦s s❡❣✉✐♥t❡s ✐t❡♥s✿
✭✐✮ Gpi ={gpi|g ∈G}❀
✭✐✐✮ Ωi(G) ={g ∈G|gp
i = 1}❀
✭✐✐✐✮ |G: Ωi(G)|=|Gp
i
|✳
◆♦ ❡♥t❛♥t♦✱ s❛❜❡♠♦s q✉❡ ♥ã♦ ❛♣❡♥❛s ♦sp✲❣r✉♣♦s ❛❜❡❧✐❛♥♦s s❛t✐s❢❛③❡♠ ❡ss❛s ❝♦♥❞✐çõ❡s✳ ❊♠ ❬✽❪✱ p✲❣r✉♣♦s s❛t✐s❢❛③❡♥❞♦ ❡ss❡s três ✐t❡♥s ❢♦r❛♠ ❞❡♥♦♠✐♥❛❞♦s ❞❡ ♣♦✇❡r ❛❜❡❧✐❛♥✳
❉❛❞♦ G ✉♠ ❣r✉♣♦ ❡ ❡❧❡♠❡♥t♦s x, y ∈ G✳ ❆ ❋ór♠✉❧❛ ❞❡ P❤✐❧✐♣ ❍❛❧❧ ❞✐③ q✉❡ ❡①✐st❡♠ ❡❧❡♠❡♥t♦s ci(x, y)∈γi(hx, yi) t❛✐s q✉❡
(xy)n =xnync2(x, y)(
n
2)c3(x, y)(
n
3). . . cn−1(x, y)(
n
n−1)cn(x, y)
♣❛r❛ t♦❞♦ n ∈ N✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥♦ ❆♣ê♥❞✐❝❡ ❆ ❞❡ ❬✹❪✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ss❛
❢ór♠✉❧❛ é q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s n = p ❡ ✐ss♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡ (xy)p = xpypzc p✱
♦♥❞❡ z ∈ γ2(hx, yi)p ❡ cp = cp(x, y) ∈ γp(hx, yi)✳ ❯♠ p✲❣r✉♣♦ ✜♥✐t♦ é ❞✐t♦ s❡r r❡❣✉❧❛r
q✉❛♥❞♦ cp(x, y)∈γ2(hx, yi)p✳ ❖s p✲❣r✉♣♦s r❡❣✉❧❛r❡s sã♦ ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s q✉❡ ♥ã♦ sã♦
♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛❜❡❧✐❛♥♦s ♠❛s ♣♦ss✉❡♠ ❛ ❡str✉t✉r❛ ♣♦✇❡r ❛❜❡❧✐❛♥✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ❛tr❛✈és ❞♦ ❚❡♦r❡♠❛ ✷✳✶✵ ❞❡ ❬✺❪✳
❊♠ ❬✷✵❪✱ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✾✽✼✱ ❆✳ ▲♦❜♦t③❦② ❡ ❆✳ ▼❛♥♥ ❞❡s❡♥✈♦❧✈❡r❛♠ ❛ t❡♦r✐❛ s♦❜r❡ p✲❣r✉♣♦s ✜♥✐t♦s ♣♦✇❡r❢✉❧✳ ❉✐③❡♠♦s q✉❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ é ♣♦✇❡r❢✉❧ s❡[G, G]6G4✱ ♣❛r❛
p = 2✱ ♦✉ [G, G] 6 Gp✱ ♣❛r❛ p í♠♣❛r✳ ❊❧❡s ♦❜s❡r✈❛r❛♠ q✉❡ ❛ ❡str✉t✉r❛ ❞❡ss❡s p✲❣r✉♣♦s
é ❜❛st❛♥t❡ s❡♠❡❧❤❛♥t❡ à ❞♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ❢♦✐ ♣r♦✈❛❞♦ q✉❡ Gpi é ♣r❡❝✐s❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s pi✲♣♦tê♥❝✐❛s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ❡♠
✷✵✵✷✱ ▲✳ ❲✐❧s♦♥ ❬✷✸❪✱ ❞❡♠♦♥str♦✉ ❡♠ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ q✉❡ q✉❛♥❞♦pé í♠♣❛r✱ Ωi(G)
■♥tr♦❞✉çã♦ ✺
❍ét❤❡❧②✐ ❡ ▲✳ ▲é✈❛✐ ❬✶✷❪ ♣r♦✈❛r❛♠ q✉❡ |Ω1(G)| = |G : Gp|✳ ❖ q✉❡ s❡r✐❛ ♦ ú❧t✐♠♦ ♣❛ss♦
♣❛r❛ ✈❡r✐✜❝❛r q✉❡ p✲❣r✉♣♦s ♣♦✇❡r❢✉❧ sã♦ ♣♦✇❡r ❛❜❡❧✐❛♥✳
❊♠ ❬✶❪✱ ❉✳ ❆r❣❛♥❜r✐❣❤t ♠♦str♦✉ q✉❡ s❡ G é ✉♠ p✲❣r✉♣♦✱ ❝♦♠ p í♠♣❛r✱ q✉❡ s❛t✐s❢❛③ γp−1(G)6Gp✱ ❡♥tã♦ Gp é ♦ ❝♦♥❥✉♥t♦ ❞❛sp✲és✐♠❛s ♣♦tê♥❝✐❛s ❞❡G✳ ■ss♦ ❧❡✈♦✉ à ❞❡✜♥✐çã♦
❞♦s p✲❣r✉♣♦s ♣♦t❡♥t✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦s p✲❣r✉♣♦s ♣♦✇❡r❢✉❧✳ ❉✐③❡♠♦s q✉❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ é ♣♦t❡♥t s❡ [G, G] 6 G4 ♣❛r❛ p = 2 ♦✉ γ
p−1(G) 6 Gp✱
♣❛r❛ p > 2✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ p = 2 ❡ p = 3 s❡r ♣♦t❡♥t é ♦ ♠❡s♠♦ q✉❡ s❡r ♣♦✇❡r❢✉❧✳ ❊♠ ❣❡r❛❧✱ q✉❛❧q✉❡r p✲❣r✉♣♦ ♣♦✇❡r❢✉❧ é t❛♠❜é♠ ♣♦t❡♥t✳ ❆ ❡str✉t✉r❛ ❞♦s p✲❣r✉♣♦s ♣♦t❡♥t ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❏✳ ●♦♥③á❧❡③✲❙á♥❝❤❡③ ❡ ❆✳ ❏❛✐❦✐♥✲❩❛♣✐r❛✐♥ ❡♠ ❬✽❪ ❡ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ❡❧❡s ♦❜t✐✈❡r❛♠ ❡stã♦ r❡✉♥✐❞♦s ♥♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳
❚❡♦r❡♠❛ ✭❆✮✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ♣♦t❡♥t✳
(i) ❙❡ p= 2✱ ❡♥tã♦✿
(a) ❖ ❡①♣♦❡♥t❡ ❞❡Ωi(G)é ♥♦ ♠á①✐♠♦2i+1 ❡✱ ♠❛✐s ❛✐♥❞❛✱[Ωi(G), G]2
i
= Ωi(G2)2
i = 1❀
(b) ❆ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ Ωi(G) é ♥♦ ♠á①✐♠♦ ⌊(i+ 2)/2⌋❀
(c) ❙❡ N EG ❡ N 6G2 ❡♥tã♦ N é ♣♦✇❡r ❛❜❡❧✐❛♥❀
(d) ❙❡ N EG ❡ N 6G4 ❡♥tã♦ N é ♣♦✇❡r❢✉❧✳
(ii) ❙❡ p >2✱ ❡♥tã♦✿
(a) ❖ ❡①♣♦❡♥t❡ ❞❡ Ωi(G) é ♥♦ ♠á①✐♠♦ pi❀
(b) ❆ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ Ωi(G) é ♥♦ ♠á①✐♠♦ (p−2)i+ 1❀
(c) ❙❡ N EG ❡♥tã♦ N é ♣♦✇❡r ❛❜❡❧✐❛♥❀
(d) ❙❡ N EG ❡ N 6Gp ❡♥tã♦ N é ♣♦✇❡r❢✉❧✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ p í♠♣❛r✱ ✈❡♠♦s q✉❡ ✉♠ p✲❣r✉♣♦ ♣♦t❡♥t é ♣♦✇❡r ❛❜❡❧✐❛♥✳
❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ❡ d(G) ❛ q✉❛♥t✐❞❛❞❡ ♠í♥✐♠❛ ❞❡ ❣❡r❛❞♦r❡s ❞❡ G✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❇❛s❡ ❞❡ ❇✉r♥s✐❞❡✱ ❚❡♦r❡♠❛ ✶✳✻ ❞❡ ❬✺❪✱ t❡♠♦s q✉❡ |G: Φ(G)|=pd(G)✳ ❈♦♥s✐✲ ❞❡r❛♥❞♦ G ❛❜❡❧✐❛♥♦ ✐ss♦ s✐❣♥✐✜❝❛ |G:Gp|=pd(G) ❡ ❛ss✐♠ t❡♠♦s d(G) = log
p(|G:Gp|) =
logp(|Ω1(G)|)✳ ❉❡ss❛ ❢♦r♠❛ é ❞❡ s❡ ❡s♣❡r❛r ❛ ♣❡r❣✉♥t❛ s❡ ❡♠ ♣♦✇❡r❢✉❧ ✐ss♦ ❛✐♥❞❛ s❡r✐❛
✈á❧✐❞♦✳ ◆❡ss❡ s❡♥t✐❞♦ ❇✳ ❑❧♦♣s❝❤ ❡ ■✳ ❙♥♦♣❝❡✱ ❬✶✼❪✱ q✉❡st✐♦♥❛r❛♠ s❡✱ ♣❛r❛ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ G❡p✉♠ ♣r✐♠♦ í♠♣❛r✱d(G) = logp(|Ω1(G)|)é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡
■♥tr♦❞✉çã♦ ✻
❚❡♦r❡♠❛ ✭❇✮✳ ❙❡❥❛♠ p ≥ 5 ❡ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿
(i) G é ♣♦✇❡r❢✉❧
(ii) d(G) = logp(|Ω1(G)|)✳
❈♦♠♦ ❝✐t❛♠♦s ❛❝✐♠❛ ❡♠ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ G✱ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s ❝♦✐♥✲ ❝✐❞❡ ❝♦♠ logp(|G : Φ(G)|) = logp(|G : Gp[G, G]|)✱ ❧❡♠❜r❛♥❞♦ q✉❡ ❡♠ p✲❣r✉♣♦s ✜♥✐t♦s
Φ(G) = Gp[G, G]✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ❚❡♦r❡♠❛ ✭❇✮ ❞✐③❡♥❞♦ q✉❡ |Ω
1(G)| =
|G:Gp[G, G]|é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠p✲❣r✉♣♦Gs❡r ♣♦✇❡r❢✉❧✳ ❊s✲
❝r❡✈❡♥❞♦ ❞❡ss❛ ❢♦r♠❛✱ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r s❡ ♠♦str❛ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✭❇✮✱ ♦ q✉❛❧ t❛♠❜é♠ ✐♥❝❧✉✐ ♦ ❝❛s♦ ❡♠ q✉❡ p= 3✳
❚❡♦r❡♠❛ ✭❈✮✳ ❙❡❥❛♠ p ✉♠ ♣r✐♠♦ í♠♣❛r✱ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ❡ s❡❥❛ k ≤p−2 ❡ i≥ 1 ♦✉ k =p−1 ❡ i≥2✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿
(i) γk(G)6Gp
i ✳
(ii) |G:Gpi
γk(G)|=|Ω{i}(G)|✳
❖ ❚❡♦r❡♠❛ ✭❇✮ ❢♦✐ ❞❡♠♦♥str❛❞♦ ❡♠ ❬✶✵❪ ❝♦♠♦ ❝♦r♦❧ár✐♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r q✉❛♥❞♦ k = 2 ❡i= 1✱ ♣❛r❛ p≥5✳ ◗✉❛♥❞♦k =p−1❡i= 1 ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✭❈✮ ♥ã♦ é s❛t✐s❢❡✐t❛✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ❛tr❛✈és ❞♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r✳ ▲❡♠❜r❡✲s❡ q✉❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ G ❞❡ ♦r❞❡♠ps✱ ♣❛r❛ ❛❧❣✉♠ s ∈ N✱ é ❞✐t♦ s❡r ❞❡ ❝❧❛ss❡ ♠❛①✐♠❛❧ s❡ G ♣♦ss✉✐ ❝❧❛ss❡
❞❡ ♥✐❧♣♦tê♥❝✐❛ ✐❣✉❛❧ ❛ s−1✳
❚❡♦r❡♠❛ ✭❉✮✳ ❙❡❥❛♠ G✉♠p✲❣r✉♣♦✱ ❝♦♠p✉♠ ♣r✐♠♦ í♠♣❛r ❡ s✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❝♦♠ s ≥p+ 1✳ ❊♥tã♦ ❡①✐st❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ t❛❧ q✉❡✿
(i) |G|=ps✳
(ii) G é ❞❡ ❝❧❛ss❡ ♠❛①✐♠❛❧✳
(iii) |G:Gpγ
p−1(G)|=|Ω1(G)|✳
(iv) γp−1(G)Gp✳
❖ t❡♦r❡♠❛ ❛♥t❡r✐♦r ♠♦str❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❡ ♣❛r❛p= 3 ♦ ❚❡♦r❡♠❛ ✭❇✮ ♥ã♦ é ✈á❧✐❞♦✱ ♣♦✐s γp−1(G) = γ2(G)G✱ ♦✉ s❡❥❛✱ G ♥ã♦ s❡r✐❛ ♣♦✇❡r❢✉❧✳
■♥tr♦❞✉çã♦ ✼
✷ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦s p✲❣r✉♣♦s r❡❣✉❧❛r❡s✱ ❞❡ ❝❧❛ss❡ ♠❛①✐♠❛❧ ♣♦✇❡r❢✉❧ ❡ ♣♦t❡♥t✳
◆♦ ❈❛♣ít✉❧♦ ✸ ♠♦str❛r❡♠♦s ❛ ❡str✉t✉r❛ ❞♦s s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ♣♦t❡♥t ❡ ❞❡♠♦♥str❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ✭❆✮✱ q✉❡ ❡st✉❞❛♠♦s ❛tr❛✈és ❞♦ ❛rt✐❣♦ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ ♥♦r♠❛❧ s✉❜❣r♦✉♣s ♦❢ ♣♦t❡♥t ♣✲❣r♦✉♣s ❞❡ ❏✳ ●♦♥③á❧❡③✲❙á♥❝❤❡③ ❡ ❆✳ ❏❛✐❦✐♥✲❩❛♣✐r❛✐♥✳ ❊ss❡ ❛rt✐❣♦ ✐♥tr♦❞✉③✐✉ ♦ ❝♦♥❝❡✐t♦ ❞❡p✲❣r✉♣♦ ♣♦t❡♥t✱ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♥❛ t❡♦r✐❛ ❞♦sp✲❣r✉♣♦s ✜♥✐t♦s✳
◆♦ ❈❛♣ít✉❧♦ ✹ ♣r♦✈❛r❡♠♦s ♦s ❚❡♦r❡♠❛s ✭❇✮ ❡ ✭❈✮✳ P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✺ ❝♦♥str✉✐r❡♠♦s ❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥tr❛❡①❡♠♣❧♦s q✉❡ ❞❡♠♦♥str❛♠ ♦ ❚❡♦r❡♠❛ ✭❉✮✳ ❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡ss❡s ú❧t✐♠♦s ❞♦✐s ❝❛♣ít✉❧♦s sã♦ ❞♦ ❛rt✐❣♦ ❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♣♦✇❡r❢✉❧ ♣✲❣r♦✉♣s ❞❡ ❏✳ ●♦♥③á❧❡③✲❙á♥❝❤❡③ ❡ ❆✳ ❩✉❣❛❞✐✲❘❡✐③❛❜❛❧✳ ❊ss❡ ❛rt✐❣♦ é ♠✉✐t♦ ✐♥t❡r❡ss❛♥t❡ ♣♦✐s ♥❡❧❡ t♦❞❛s ❛s ❢❛♠í❧✐❛s ❞❡ p✲❣r✉♣♦s ✜♥✐t♦s ❞❡✜♥✐❞❛s ♥♦ ❈❛♣ít✉❧♦ ✷ sã♦ ✉t✐❧✐③❛❞❛s ❡ r❡❧❛❝✐♦♥❛❞❛s✳
❈❛♣ít✉❧♦
1
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❚❡♦r✐❛ ❞❡ ●r✉✲ ♣♦s ❡ ▼ó❞✉❧♦s q✉❡ ✉s❛r❡♠♦s ♥♦ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❖♠✐t✐r❡♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧✲ t❛❞♦s ❛q✉✐ ❛♣r❡s❡♥t❛❞♦s✱ ♠❛s t♦❞♦s ❡❧❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♥❛s r❡❢❡rê♥❝✐❛s ❝✐t❛❞❛s✳
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s
❖ ❡st✉❞♦ s♦❜r❡ ❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❢❡✐t❛ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❜❛s❡❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ❧✐✈r♦s ❋✐♥✐t❡ ●r♦✉♣s ❬✶✶❪✱ ❆❧❣❡❜r❛✲❆ ●r❛❞✉❛t❡ ❈♦✉rs❡ ❬✶✺❪✱ ❆♥❛❧②t✐❝ Pr♦✲♣ ●r♦✉♣s ❬✹❪✱ ❡ ♥♦ ❛rt✐❣♦ ❬✺❪✳ ◆♦ss♦ ❡st✉❞♦ é ❜❛st❛♥t❡ s✉❝✐♥t♦✱ ♣♦r ✐ss♦ ❛ss✉♠✐r❡♠♦s ❝♦♠♦ ❝♦♥❤❡❝✐❞♦s ♠✉✐t♦s r❡s✉❧t❛❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦s ❚❡♦r❡♠❛s ❞♦ ■s♦♠♦r✜s♠♦ ❡ ♦ ❞❛ ❈♦rr❡s♣♦♥❞ê♥❝✐❛✱ ❞❡♥tr❡ ♦✉tr♦s✳
❉❛❞♦ ✉♠ ❣r✉♣♦ G✱ ❞❡♥♦t❛r❡♠♦s o(g)♣❡❧❛ ♦r❞❡♠ ❞♦ ❡❧❡♠❡♥t♦ g ∈G✱|G|✱ ❝♦♠♦ s❡♥❞♦
❛ ♦r❞❡♠ ❞♦ ❣r✉♣♦ G✱|G:H|✱ ♦ í♥❞✐❝❡ ❞♦ s✉❜❣r✉♣♦H ♥♦ ❣r✉♣♦G ❡ Z(G)♦ ❝❡♥tr♦ ❞❡ss❡ ❣r✉♣♦✳ ❉❡♠❛✐s ♥♦t❛çõ❡s s❡rã♦ ❞❡✜♥✐❞❛s ♥♦ s❡✉ ❞❡✈✐❞♦ t❡♠♣♦✳
✶✳✶✳✶ ❈♦♠✉t❛❞♦r❡s ❡ s✉❜❣r✉♣♦s ❣❡r❛❞♦s ♣♦r ❝♦♠✉t❛❞♦r❡s
❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❖ ❝♦♠✉t❛❞♦r ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s x ❡ y é ❞❡✜♥✐❞♦ ♣♦r [x, y] = x−1y−1xy✳ ❈♦♠ ✐ss♦ t❡♠♦s q✉❡x ❡ y ❝♦♠✉t❛♠ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ [x, y] = 1✳
P♦❞❡♠♦s ❞❡✜♥✐r ❝♦♠✉t❛❞♦r ❞❡ q✉❛❧q✉❡r ❝♦♠♣r✐♠❡♥t♦ ♥❛t✉r❛❧ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
[x1, . . . , xn] = [[x1, . . . , xn−1], xn],
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✾
❚❡♦r❡♠❛ ✶✳✶✳✶ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✼✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ ❝♦♥s✐❞❡r❡ x, y ❡ z ❡❧❡♠❡♥t♦s ❞❡ G✳ ❊♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♠✉t❛❞♦r❡s✿
(i) [x, y] =x−1xy.
(ii) [y, x] = [x, y]−1.
(iii) [xy, z] = [x, z]y[y, z] = [x, z][x, z, y][y, z].
(iv) [x, yz] = [x, z][x, y]z = [x, z][x, y][x, y, z].
(v) [x, y−1, z]y[y, z−1, x]z[z, x−1, y] = 1 ✭■❞❡♥t✐❞❛❞❡ ❞❡ ❍❛❧❧✲❲✐tt✮✳
(vi) yx =xy[y, x].
❚❡♦r❡♠❛ ✶✳✶✳✷ ✭❬✹❪✱ Pá❣✐♥❛ ✶ ❡ ✷✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ x, y ❡❧❡♠❡♥t♦s ❞❡ G✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n ✈❛❧❡ q✉❡✿
(i) [xn, y] = [x, y]xn−1
[x, y]xn−2
· · ·[x, y]x[x, y]✳
(ii) [x, yn] = [x, y][x, y]y· · ·[x, y]yn−1
✳
❙❡❥❛♠ H ❡ K s✉❜❣r✉♣♦s ❞❡ ✉♠ ❣r✉♣♦ G✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ s✉❜❣r✉♣♦
❝♦♠✉t❛❞♦r ❞❡ H ❡ K ♣♦r [H, K] = h[h, k] | h ∈ H, k ∈ Ki✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱
❞❡✜♥✐♠♦s ♦ s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r ❞❡ q✉❛❧q✉❡r ❝♦♠♣r✐♠❡♥t♦ ♥❛t✉r❛❧ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ [H1, . . . , Hn] = [[H1, . . . , Hn−1], Hn]✱ ♦♥❞❡ H1, . . . , Hn sã♦ s✉❜❣r✉♣♦s ❞❡G✳
❚❡♦r❡♠❛ ✶✳✶✳✸ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✼✮✳ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ H, K ❡ L s✉❜❣r✉♣♦s ●✳ ❊♥tã♦✿
(i) [H, K] = [K, H]✳
(ii) H 6NG(K) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ [H, K]6K✳
(iii) H 6CG(K) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ [H, K] = 1✳
(iv) [H, K]σ = [Hσ, Kσ]✱ ♣❛r❛ q✉❛❧q✉❡r ❡♥❞♦♠♦r✜s♠♦ σ : G → G✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦
s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r ❞❡ ❞♦✐s s✉❜❣r✉♣♦s ❝❛r❛❝t❡ríst✐❝♦s ✭♥♦r♠❛✐s✮ ❞❡ G é ❛✐♥❞❛ é ✉♠ s✉❜❣r✉♣♦ ❝❛r❛❝t❡ríst✐❝♦ ✭♥♦r♠❛❧✮✳
(v) ❙❡ N é ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ ●✱ ❡♥tã♦ [HN/N, KN/N] = [H, K]N/N✳
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✶✵
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ♠✉✐t♦ út✐❧ q✉❛♥❞♦ s❡ tr❛t❛ ❞♦ s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r✳ P♦✐s✱ ❞❛❞♦ ✉♠ ❣r✉♣♦ G ❡ q✉❛tr♦ s✉❜❣r✉♣♦s ❞❡❧❡✱ s❡♥❞♦ ✉♠ ❞❡❧❡s ♥♦r♠❛❧✱ ♦ ❧❡♠❛ ♥♦s ♠♦str❛ ✉♠❛ r❡❧❛çã♦ ❞❡ ♣❡rt✐♥ê♥❝✐❛ ❡♥tr❡ ♦ s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r ❞❡ss❡s s✉❜❣r✉♣♦s ❡ ♦ ♥♦r♠❛❧✳ ❊ss❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡♠❛ ❞♦s ❚rês ❙✉❜❣r✉♣♦s✳
▲❡♠❛ ✶✳✶✳✹ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✽✮✳ ✭▲❡♠❛ ❞♦s ❚rês ❙✉❜❣r✉♣♦s✮❙❡❥❛ G✉♠ ❣r✉♣♦✱ H, J ❡ K s✉❜❣r✉♣♦s ❞❡G ❡ N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ Gt❛❧ q✉❡ [H, J, K],[K, H, J]6N✳ ❊♥tã♦ [J, K, H]6N✳
❊♠ ♥♦ss♦ tr❛❜❛❧❤♦ t❛♠❜é♠ ✉s❛r❡♠♦s ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ss❡ t❡♦r❡♠❛✱ ♦♥❞❡ s❡ ❝♦♥✲ s✐❞❡r❛r♠♦s H = L ❡ J = N = K✱ ❝♦♠ L ❡ N s✉❜❣r✉♣♦s ♥♦r♠❛✐s✱ ❡♥tã♦ [N, N, L] 6 [L, N, N]✳
❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠✉t❛❞♦r❡s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s✉❜❣r✉♣♦s ❞❡ ✉♠ ❣r✉♣♦ G✱ ❞❡♥♦♠✐♥❛❞❛ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r ❞❡ G✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
γ1(G) =G, γ2(G) = [G, G] =G′, γn(G) = [γn−1(G), G] ♣❛r❛ n >2.
❉♦ ♠♦❞♦ ❝♦♠♦ ❡ss❛ sér✐❡ é ❞❡✜♥✐❞❛✱ ❢❛❝✐❧♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❝❛❞❛ t❡r♠♦ ❞❡❧❛ é ❝❛r❛❝t❡ríst✐❝♦ ❡♠G✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❛ s❛t✐s❢❛③γi+1(G)6γi(G)✱ ❡ ✐ss♦ ❛❝❛rr❡t❛ q✉❡ ❛ sér✐❡
é ❝❡♥tr❛❧ ❡♠ G✱ ♦✉ s❡❥❛✱ γi(G)/γi+1(G)6Z(G/γi+1(G))✳
❖s ♣ró①✐♠♦s ❞♦✐s t❡♦r❡♠❛s sã♦ ❛♣❧✐❝❛çõ❡s ❞♦ ▲❡♠❛ ❞♦s ❚rês ❙✉❜❣r✉♣♦s✱ ♦♥❞❡ ♦ ♣r✐✲ ♠❡✐r♦ ♥♦s ♠♦str❛ ✉♠❛ r❡❧❛çã♦ ♠✉✐t♦ út✐❧ ❞❛ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r✳
❚❡♦r❡♠❛ ✶✳✶✳✺ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✾✮✳ P❛r❛ q✉❛❧q✉❡r ❣r✉♣♦ G✱ [γi(G), γj(G)] 6 γi+j(G)✱
♣❛r❛ t♦❞♦ i, j ≥1✳
❚❡♦r❡♠❛ ✶✳✶✳✻ ✭❬✶✻❪✱ ▲❡♠❛ ✹✳✾✮✳ ❙❡❥❛ G✉♠ ❣r✉♣♦ ❡N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡G✳ ❊♥tã♦ [γk(N), G]6[N, γk(G)]✱ ♣❛r❛ t♦❞♦ k≥1✳
❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r✱ ♣r♦✈❡♥✐❡♥t❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✳✸✱ ✐t❡♠ (v)✱ é ♦ ♥♦ss♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✱ q✉❡ ❞❡ ❝❡rt❛ ❢♦r♠❛ ♥♦s ♠♦str❛ ❝♦♠♦ ❞❡✈❡ s❡r ❡ss❛ sér✐❡ ♥♦ ❣r✉♣♦ q✉♦❝✐❡♥t❡ G/N✱ ♦♥❞❡N EG✳
❚❡♦r❡♠❛ ✶✳✶✳✼✳ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G✳ ❊♥tã♦ γi(G/N) =
γi(G)N/N✱ ♣❛r❛ t♦❞♦ i≥1✳
✶✳✶✳✷ ●r✉♣♦s ♥✐❧♣♦t❡♥t❡s
❉❛❞♦ G ✉♠ ❣r✉♣♦✱ ❞✐③❡♠♦s q✉❡ ❡❧❡ é ♥✐❧♣♦t❡♥t❡ s❡ ❡①✐st❡ c∈ N t❛❧ q✉❡ γc+1(G) = 1✳
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✶✶
q✉❛♥❞♦ c= 1✱ t❡♠♦s γ2(G) = 1 ❡ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ G é ❛❜❡❧✐❛♥♦✳ ❉❡ss❛ ❢♦r♠❛ ♦s ❣r✉♣♦s
♥✐❧♣♦t❡♥t❡s ❞❡ ❝❧❛ss❡ ✉♠ sã♦ ♣r❡❝✐s❛♠❡♥t❡ ♦s ❛❜❡❧✐❛♥♦s✳
❆ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ✉♠ ❣r✉♣♦ G s❡r ♥✐❧♣♦t❡♥t❡ t❛♠❜é♠ ♣♦❞❡ s❡r ❢❡✐t❛ ❡♠ t❡r♠♦s ❞❡ ♦✉tr❛ sér✐❡✱ ❞❡♥♦♠✐♥❛❞❛ sér✐❡ ❝❡♥tr❛❧ s✉♣❡r✐♦r ❞❡ G✳ ❘❡❝✉rs✐✈❛♠❡♥t❡✱ ❡❧❛ é ❞❡✜♥✐❞❛ ❞❛ ❢♦r♠❛ Z0(G) = 1✱ Z1(G) = Z(G) ❡✱ ♣❛r❛ i > 1✱ Zi(G) é ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ ❡♠ G
❞❡ Z(G/Zi−1(G))✱ ♦✉ s❡❥❛✱ s❛t✐s❢❛③ Zi+1(G)/Zi(G) = Z(G/Zi(G))✳ ❖ ♣ró①✐♠♦ t❡♦r❡♠❛
r❡❧❛❝✐♦♥❛ ❡ss❛ sér✐❡ ❝♦♠ ❛ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r ❡ t❛♠❜é♠ ❥✉st✐✜❝❛ ✉♠ ♣♦✉❝♦ ♦ ❢❛t♦ ❞❡❧❛s s❡r❡♠ ❝❤❛♠❛❞❛s ✐♥❢❡r✐♦r ❡ ❛ ♦✉tr❛ s✉♣❡r✐♦r✳
❚❡♦r❡♠❛ ✶✳✶✳✽ ✭❬✺❪✱ ▲❡♠❛ ✶✳✶✷✮✳ ❙❡❥❛G✉♠ ❣r✉♣♦ ♥✐❧♣♦t❡♥t❡ ❞❡ ❝❧❛ss❡c✳ ❊♥tã♦γc+1−i(G)6
Zi(G)✱ ♣❛r❛ t♦❞♦ 0≤i≤c✳
❯s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ♣♦❞❡♠♦s ❝❛r❛❝t❡r✐③❛r ❣r✉♣♦s ♥✐❧♣♦t❡♥t❡s ❝♦♥s✐❞❡r❛♥❞♦ ❛ sér✐❡ ❝❡♥tr❛❧ s✉♣❡r✐♦r✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳ ❊❧❡ t❛♠❜é♠ ♥♦s ♠♦str❛ q✉❡ ❛ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡✜♥✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡ é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛♠❜❛s ❛s sér✐❡s ❝❡♥tr❛✐s✱ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r✳
❚❡♦r❡♠❛ ✶✳✶✳✾ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✶✸✮✳ ❯♠ ❣r✉♣♦ G é ♥✐❧♣♦❡♥t❡ ❞❡ ❝❧❛ss❡ c s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Zc(G) = G ❡ Zc−1(G)6=G✳
●r✉♣♦s ✜♥✐t♦s ♥✐❧♣♦t❡♥t❡s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❝❛r❛❝t❡r✐③❛❞♦s ❛tr❛✈és ❞♦s s❡✉s s✉❜❣r✉✲ ♣♦s ❞❡ ❙②❧♦✇✱ s❡♠ ❞❡♣❡♥❞❡r ❞❡ ♥❡♥❤✉♠❛ sér✐❡✱ ❝♦♠♦ ♥♦s ❞✐③ ♦ ♣ró①✐♠♦ t❡♦r❡♠❛✳
❚❡♦r❡♠❛ ✶✳✶✳✶✵ ✭❬✶✶❪✱ ❚❡♦r❡♠❛ ✸✳✺✮✳ ❯♠ ❣r✉♣♦ G é ♥✐❧♣♦t❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ é ♣r♦❞✉t♦ ❞✐r❡t♦ ❞❡ s❡✉s s✉❜❣r✉♣♦s ❞❡ ❙②❧♦✇✳
✶✳✶✳✸ ❖ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐
❯♠ s✉❜❣r✉♣♦ ✐♠♣♦rt❛♥t❡✱ q✉❡ ❛✐♥❞❛ é ❝❛r❛❝t❡ríst✐❝♦✱ ❡♠ ✉♠ ❣r✉♣♦Gé ♦ ❞❡♥♦♠✐♥❛❞♦ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐✳ ❊❧❡ é ❞❡♥♦t❛❞♦ ♣♦r Φ(G) ❡ ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s s✉❜❣r✉♣♦s ♠❛①✐♠❛✐s ❞♦ G✳ ❈❛s♦ G✱ ♥ã♦ ♣♦ss✉❛ ♥❡♥❤✉♠ ♠❛①✐♠❛❧✱ ❞❡✜♥✐♠♦s Φ(G) =G✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ g ✉♠ ❡❧❡♠❡♥t♦ ❞❡ G✳ ❉✐③❡♠♦s q✉❡ g é ✉♠ ♥ã♦✲ ❣❡r❛❞♦r s❡ q✉❛♥❞♦ hX∪gi = G✱ ❡♥tã♦ t❡♠♦s q✉❡ hXi = G✱ ♣❛r❛ q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ X ⊆G✳
❈♦♠ ❡ss❛ ❞❡✜♥✐çã♦ ❞❡ ❡❧❡♠❡♥t♦s ♥ã♦ ❣❡r❛❞♦r❡s ❡♠ ✉♠ ❣r✉♣♦ ✜♥✐t♦✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ♥ã♦✲ ❣❡r❛❞♦r❡s ❞❡ G✳
❚❡♦r❡♠❛ ✶✳✶✳✶✷ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✺✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ x1, . . . , xn ∈ G✳ ❊♥tã♦
✶✳✶ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ✶✷
✶✳✶✳✹ ❋ór♠✉❧❛ ❞❡ ❈♦♠♣✐❧❛çã♦ ❞❡ ❍❛❧❧
❊♠ q✉❛❧q✉❡r ❣r✉♣♦ ❛❜❡❧✐❛♥♦ s❛❜❡♠♦s q✉❡ ✈❛❧❡ xnyn = (xy)n✱ ♠❛s ✐ss♦ ♥ã♦ é ✈á❧✐❞♦
❡♠ ❣❡r❛❧✳ ❆ ❋ór♠✉❧❛ ❞❡ ❈♦♠♣✐❧❛çã♦ ❞❡ P❤✐❧✐♣ ❍❛❧❧✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ♣♦r ❋ór♠✉❧❛ ❞❡ ❍❛❧❧✲P❡tr❡s❝✉✱ ♥♦s ❢♦r♥❡❝❡ ✉♠ s✉❜st✐t✉t♦ ♣❛r❛ ❡ss❡ ❢❛t♦ ✈á❧✐❞♦ ❡♠ q✉❛❧q✉❡r ❣r✉♣♦✳
❙❡❥❛♠ G ✉♠ ❣r✉♣♦✱ x, y ❡❧❡♠❡♥t♦s ❞❡ G✱ ❡ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦ (xy)n ❡
xnyn sã♦ ✐❣✉❛✐s ♠ó❞✉❧♦ G′✱ ❞❡ss❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r xnyn = (xy)nc✱ ♣❛r❛ ❛❧❣✉♠
c ∈ G′✳ ❆ ❢ór♠✉❧❛ ❞❡ ❝♦♠♣✐❧❛çã♦ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ c ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡
❝♦♠✉t❛❞♦r❡s✳
❚❡♦r❡♠❛ ✶✳✶✳✶✸ ✭❬✹❪✱ ❆♣ê♥❞✐❝❡ ❆✮✳ ❙❡❥❛♠ x ❡ y ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❣r✉♣♦ ❡ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦
xnyn = (xy)nc( n
2)
2 . . . c
(n i)
i . . . cnn−1cn
♦♥❞❡ ci ∈γi(G) ♣❛r❛ ❝❛❞❛ i✳
P♦❞❡♠♦s ❝♦♥str✉✐r ❝❛❞❛ ci ✐❣✉❛❧ ❛ ✉♠ ♣r♦❞✉t♦ ❞❡ ❝♦♠✉t❛❞♦r❡s ❡♠ x ❡ y ❞❡ ❝♦♠♣r✐✲
♠❡♥t♦ ♣❡❧♦ ♠❡♥♦s i✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❢ór♠✉❧❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♦♥❞❡ s❡ ❝♦♥s✐❞❡r❛ ci =ci(x, y)∈γi(hx, yi)✱ ♣❛r❛ ❝❛❞❛ i✳
❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ss❛ ❢ór♠✉❧❛ é q✉❛♥❞♦ t♦♠❛♠♦s n = pk ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦
♣♦s✐t✐✈♦ k✳ ❈♦♠♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ pk
i
é ❞✐✈✐sí✈❡❧ ♣♦r pk−j ♣❛r❛ pj ≤ i < pj+1✱
t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❢♦r♠✉❧❛çã♦✳
❚❡♦r❡♠❛ ✶✳✶✳✶✹ ✭❬✹❪✱ ▲❡♠❛ ✶✶✳✾✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ x, y ❡❧❡♠❡♥t♦s ❞❡ G✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ k≥0 t❡♠♦s
(xy)pk ≡xpkypk( ♠♦❞ γ2(L)p
k γp(L)p
k−1
γp2(L)p
k−2
γp3(L)p
k−3
· · ·γpk(L)), ♦♥❞❡ L=hx, yi✳ ❚❛♠❜é♠ t❡♠♦s q✉❡
[x, y]pk ≡[xpk, y]( ♠♦❞ γ2(M)p
k
γp(M)p
k−1
γp2(M)p
k−2
. . . γpk(M)), ♦♥❞❡ M =hx,[x, y]i✳
❯♠ ❝♦r♦❧ár✐♦ s✐♠♣❧❡s ❞❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡ss❡ t❡♦r❡♠❛ é q✉❛♥❞♦ t♦♠❛♠♦s ✉♠❛ q✉❛♥✲ t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G✳
❈♦r♦❧ár✐♦ ✶✳✶✳✶✺✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ x1, . . . , xr ❡❧❡♠❡♥t♦s ❞❡ G✳ ❊♥tã♦
(x1. . . xr)p
k
≡xp1k. . . xrpk(♠♦❞ γ2(L)p
k γp(L)p
k−1
γp2(L)p
k−2
γp3(L)p
k−3
✶✳✷ ❆♥é✐s ❡ ▼ó❞✉❧♦s ✶✸
✶✳✷ ❆♥é✐s ❡ ▼ó❞✉❧♦s
◆❡st❛ s❡çã♦ ❢❛❧❛r❡♠♦s ❞❡ ❛❧❣✉♥s ❛s♣❡❝t♦s ❜ás✐❝♦s ❞❛ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❡ ▼ó❞✉❧♦s q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✼❪✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ R ✉♠ ❝♦♥❥✉♥t♦ ♠✉♥✐❞♦ ❞❡ ❞✉❛s ♦♣❡r❛çõ❡s✿ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞❡♥♦t❛❞❛s ✉s✉❛❧♠❡♥t❡ ♣♦r ′+′ ❡ ♣❡❧❛ ❥✉st❛♣♦s✐çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉✐③❡♠♦s q✉❡ R é ✉♠
❛♥❡❧ s❡ ❞❛❞♦s r, s, t∈R ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s✿
❼ (R,+) é ✉♠ ❣r✉♣♦ ❛❜❡❧✐♥♦✱
❼ (rs)t=r(st)✱
❼ r(s+t) =rs+rt✱
❼ (s+t)r =sr+tr✳
❙❡R ♣♦ss✉✐r ✉♠ ❡❧❡♠❡♥t♦ uq✉❡ s❛t✐s❢❛③ur =r =ru✱ ♣❛r❛ t♦❞♦r∈R✱ ❡ss❡ ❡❧❡♠❡♥t♦
u é ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛❞♦ ♣♦r 1 ❡ ❞❡♥♦♠✐♥❛❞♦ ✉♥✐❞❛❞❡✳ ◆❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡R é ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✳ ❙❡ ❛❧é♠ ❞❛s q✉❛tr♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❛ ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❢♦r ❝♦♠✉t❛t✐✈❛✱ ❡♥tã♦ ❞✐③❡♠♦s R é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✳
❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛ M ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛❞✐t✐✈♦ ❡R ✉♠ ❛♥❡❧✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❛❞❛ m ∈ M ❡ r ∈ R✱ s❡❥❛ ❞❡✜♥✐❞♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ M✱ ❞❡♥♦t❛❞♦ ♣♦r mr✳ ❊♥tã♦ M é ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ s❡ ♣❛r❛ q✉❛✐sq✉❡r x, y ∈M ❡ r, s∈R ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ✈❛❧❡♠✿
❼ (x+y)r=xr+yr✱
❼ x(r+s) = xr+xs✱
❼ x(rs) = (xr)s✱
❼ x1 = x✳
❚❛♠❜é♠ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ ♦♥❞❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ M q✉❡ ❛❣♦r❛ é ❞❡♥♦t❛❞♦ ♣♦r rm ❡ s❛t✐s❢❛③ ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s às ❝✐t❛❞❛s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳
❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❯♠ R✲s✉❜♠ó❞✉❧♦ N ❞❡ ✉♠ R✲♠ó❞✉❧♦ M é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦
♣❛r❛ t♦❞❛s ❛s ♦♣❡r❛çõ❡s ❞❡ ♠ó❞✉❧♦✱ ♦✉ s❡❥❛✱ N é ✉♠ s✉❜❣r✉♣♦ ❛❞✐t✐✈♦ ❞❡ M ❡ nr ∈ N✱
✶✳✷ ❆♥é✐s ❡ ▼ó❞✉❧♦s ✶✹
❙❡❥❛ M ✉♠ R✲♠ó❞✉❧♦✱ s❡❥❛♠ t ∈ N ❡ m1, . . . , mt ❡❧❡♠❡♥t♦s ❞❡ M✳ ❈♦♥s✐❞❡r❛♠♦s ♦
s❡❣✉✐♥t❡ s✉❜❝♦♥❥✉♥t♦ N ❞❡ M✿
N =m1R+· · ·+mtR={m1r1+· · ·+mtrt | ri ∈R}.
❖ ❝♦♥❥✉♥t♦Né ✉♠ s✉❜♠ó❞✉❧♦ ❞❡M ❡ é ❞❡♥♦♠✐♥❛❞♦ s✉❜♠ó❞✉❧♦ ❣❡r❛❞♦ ♣♦rm1, . . . , mt✳
❉✐③❡♠♦s q✉❡M é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦sm1, . . . , mt
❞❡ M t❛✐s q✉❡
M =m1R+· · ·+mtR.
◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡m1, . . . , mt é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ ♦ ♠ó❞✉❧♦ M✳
❯♠R✲♠ó❞✉❧♦ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ♥❡ss❡ tr❛❜❛❧❤♦ s❡rá ❞❡♥♦t❛❞♦ ♣♦rRt✱
♦♥❞❡ t ∈N✳ ❉❡✜♥✐♠♦sRt ={(r
1, . . . , rt)| ri ∈R}❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❝♦♦r❞❡♥❛❞❛
❛ ❝♦♦r❞❡♥❛❞❛ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❛♥❡❧ ❛t✉❛ ❡♠ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛✱ ♦✉ s❡❥❛✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
(r1, . . . , rt) + (r′1, . . . , r′t) := (r1+r′1, . . . , rt+rt′)
❡
r(r1, . . . , rt) := (rr1, . . . , rrt).
❖ ❝♦♥❥✉♥t♦ Rt é ✉♠ R✲♠ó❞✉❧♦ ❣❡r❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s e
1, . . . , et✱ ♦♥❞❡ ❝❛❞❛ ei =
(0, . . . ,1i, . . . ,0)❝♦♠ i∈ {1, . . . , t}✳ ❖✉ s❡❥❛✱ Rt é ✉♠ R✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳
❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ M ✉♠ R✲♠ó❞✉❧♦✳ ❙❡❥❛ N ✉♠ R✲s✉❜♠ó❞✉❧♦ ❞❡
M✳ ❊♥tã♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ (N,+) é ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ (M,+) ❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❣r✉♣♦ q✉♦❝✐❡♥t❡ (M/N,+N)✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ {m+N | m∈ M} ❞❛s ❝❧❛ss❡s ❧❛t❡r❛✐s
❞❡ N ❡♠ M ♠✉♥✐❞♦ ❞❛ ❛❞✐çã♦
+N :M/N ×M/N −→M/N
(m1+N, m2+N)7−→(m1+m2) +N.
❙♦❜r❡ ❡st❡ ❣r✉♣♦ (M/N,+N)✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ s❡❣✉✐♥t❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r
❡♠ R
R×M/N −→M/N (r, m+N)7−→mr+N.
✶✳✷ ❆♥é✐s ❡ ▼ó❞✉❧♦s ✶✺
❝✐❡♥t❡ ❞❡ M ♣♦r N✳
❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❙❡❥❛ M ✉♠ R✲♠ó❞✉❧♦✳ ❉✐③❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s m1, . . . , mt ❞❡ M
sã♦ R✲❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s q✉❛♥❞♦✱ ❞❛❞♦s ri ∈R✱♣❛r❛ t♦❞♦ i ✈❛❧❡ t
X
i=1
rimi = 0⇒ri = 0.
❯♠R✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ M é ❞✐t♦ s❡r ❧✐✈r❡ s❡ ❡❧❡ ❛❞♠✐t❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ❣❡r❛❞♦r❡s m1, . . . , mt q✉❡ sã♦R✲❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ s❡ ♦
♠ó❞✉❧♦ M é ✐s♦♠♦r❢♦ ❛ Rt✳
❊ss❛ ❡q✉✐✈❛❧ê♥❝✐❛ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ♦ ❤♦♠♦♠♦r✜s♠♦
ϕ :Rt−→M (r1, . . . , rt)7−→
t
X
i=1
miri.
◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ {m1, . . . , mt} é ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ♠ó❞✉❧♦ ❧✐✈r❡ M✳ ❖❜s❡r✈❡
❛✐♥❞❛ q✉❡ M =m1R⊕ · · · ⊕mtR✳
❚❡♦r❡♠❛ ✶✳✷✳✻✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡ ❡ M ✉♠ R✲♠ó❞✉❧♦ ❧✐✈r❡ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦✳ ❊♥tã♦✱ t♦❞❛s ❛s ❜❛s❡s ❞❡ M ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✳
❈♦♠ ❡ss❡ t❡♦r❡♠❛ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣❡r❛❞♦r❡s ❞❡ ✉♠R✲♠ó❞✉❧♦ ❧✐✈r❡ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❡stá ❜❡♠✲❞❡✜♥✐❞❛✱ ♦ q✉❡ é ♥♦r♠❛❧♠❡♥t❡ ❞❡♥♦♠✐♥❛❞♦ ♣♦st♦ ❞❡ M ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r dR(M)✳
❆ ♣ró①✐♠❛ ❡ ú❧t✐♠❛ ❞❡✜♥✐çã♦ ❞❡ss❛ s❡çã♦ ❢♦✐ r❡t✐r❛❞♦ ❞♦ ❧✐✈r♦ ❚❤❡ ❙tr✉❝t✉r❡ ♦❢ ●r♦✉♣s ♦❢ Pr✐♠❡ P♦✇❡r ❖r❞❡r✱ ❬✶✾✱ ❈❛♣ít✉❧♦ ✹✱ Pá❣✐♥❛ ✾✸❪✳ ❊❧❛ s❡rá ✉t✐❧✐③❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✺✱ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ p✲❣r✉♣♦s ✜♥✐t♦s✳
❈❛♣ít✉❧♦
2
❆❧❣✉♠❛s ❢❛♠í❧✐❛s ❞❡
p
✲❣r✉♣♦s ✜♥✐t♦s
❯♠ ❣r✉♣♦G♥♦ q✉❛❧ t♦❞♦ ❡❧❡♠❡♥t♦ t❡♠ ❝♦♠♦ ♦r❞❡♠ ✉♠❛ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ❝❡rt♦ ♣r✐♠♦ p é ❞✐t♦ s❡r ✉♠ p✲❣r✉♣♦✳ ◗✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦ G✱ ❡ss❛ ❞❡✜♥✐çã♦ s✐❣♥✐✜❝❛ q✉❡ |G|=pn✱ ♣❛r❛ ❛❧❣✉♠n ∈N✳
◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ✈✐♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ♣❛r❛ ❣r✉♣♦s q✉❛✐sq✉❡r✱ ❛❣♦r❛ ❛♣r♦✲ ❢✉♥❞❛r❡♠♦s ♥♦ss♦ ❡st✉❞♦ ❡♠ p✲❣r✉♣♦s ✜♥✐t♦s✳ ❈♦♠ ❡ss❡ ✐♥t✉✐t♦✱ ♣r✐♠❡✐r♦ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ♣r❡❧✐♠✐♥❛r s♦❜r❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛❧❣✉♥s ❞❡ s❡✉s s✉❜❣r✉♣♦s✱ ♠✉✐t♦ ✉t✐❧✐③❛❞♦s ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❊♠ s❡❣✉✐❞❛ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ❢❛♠í❧✐❛s✱ ❜❡♠ ❝♦♠♦ ❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ❝❛❞❛ ✉♠❛✳
❆ ♣❛rt✐r ❞❛q✉✐✱ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s ♦♠✐t✐r❡♠♦s ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞♦p✲❣r✉♣♦ s❡r ✜♥✐t♦✱ ♠❛s t❡♥❤❛ ❡♠ ♠❡♥t❡ q✉❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ✐ss♦✳
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s
◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ❞❡✜♥✐♠♦s ✉♠ ❣r✉♣♦ ♥✐❧♣♦t❡♥t❡✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛✱ ♠✉✐t♦ ✐♠♣♦r✲ t❛♥t❡ ❞❡ ✉♠p✲❣r✉♣♦ ✜♥✐t♦ é q✉❡ ❡❧❡s sã♦ ♥✐❧♣♦t❡♥t❡s✳ ❆❧é♠ ❞✐ss♦✱ ✈❛❧❡ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✱ q✉❡ ♥♦s ♠♦str❛✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ ❛ s✉❛ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛✳
❚❡♦r❡♠❛ ✷✳✶✳✶ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✶✺✮✳ ❙❡❥❛ G ✉♠p✲❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠pm ≥p2✳ ❊♥tã♦✿
(i) ❆ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ G é✱ ♥♦ ♠á①✐♠♦✱ m−1✳
(ii) ❙❡ G t❡♠ ❝❧❛ss❡ ❞❡ ♥✐❧♣♦tê♥❝✐❛ c✱ ❡♥tã♦ |G:Zc−1(G)| ≥p2✳
(iii) |G:G′| ≥p2✳
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✶✼
❈♦r♦❧ár✐♦ ✷✳✶✳✷ ✭❬✺❪✱ ❈♦r♦❧ár✐♦ ✶✳✶✻✮✳ ❙❡❥❛♠ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ❡ N ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G ❝♦♠ í♥❞✐❝❡ pi ≥p2✳ ❊♥tã♦ γ
i(G)6N✳
❊♠ ✉♠p✲❣r✉♣♦ ✜♥✐t♦ ♦ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐✱ ❞❡✜♥✐❞♦ ♥❛ ❙❡çã♦ ✶ ❞♦ ❈❛♣ít✉❧♦ ✶✱ ♣♦ss✉✐ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♠✉✐t♦ út✐❧ ❞❡♥tr♦ ❞❡ss❛ t❡♦r✐❛✱ ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳
❚❡♦r❡♠❛ ✷✳✶✳✸ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✷✳✷✮✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ ❊♥tã♦ Φ(G) =Gp[G, G]✳
❖✉tr❛ ✐♥❢♦r♠❛çã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ q✉❡ ♦ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ ♥♦s ♠♦str❛✱ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ p✲❣r✉♣♦s ✜♥✐t♦s✱ é s♦❜r❡ ❛ q✉❛♥t✐❞❛❞❡ ♠í♥✐♠❛ ❞❡ ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦✳ ❈♦♠♦ ✈❡r❡♠♦s ♥♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❛ ❇❛s❡ ❞❡ ❇✉r♥s✐❞❡✳ ❉❡♥♦t❛♠♦s d(G) ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ G❡ Fp ❝♦♠♦ s❡♥❞♦ ✉♠ ❝♦r♣♦
✜♥✐t♦ ❝♦♠ p ❡❧❡♠❡♥t♦s✳
❚❡♦r❡♠❛ ✷✳✶✳✹ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✶✳✻✮✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ ❊♥tã♦✿
(i) G/Φ(G) é ✉♠ p✲❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡❧❡♠❡♥t❛r ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ Fp✳
(ii) ❖ ❝♦♥❥✉♥t♦ {x1, . . . , xd} é ✉♠ ❝♦♥❥✉♥t♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ G s❡✱ ❡ s♦♠❡♥t❡
s❡✱ {x1Φ(G), . . . , xdΦ(G)} é ✉♠❛ ❜❛s❡ ♣❛r❛G/Φ(G)✳
(iii) ❖ ♥ú♠❡r♦ ♠í♥✐♠♦ d =d(G) ❞❡ ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ G ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞✐♠❡♥sã♦ ❞❡ G/Φ(G) ❝♦♠♦ ✉♠ Fp−❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ |G: Φ(G)|=pd✳
❉❡✜♥✐♠♦s ♦ ❡①♣♦❡♥t❡ ❞❡ ✉♠ ❣r✉♣♦ G✱ ❞❡♥♦t❛❞♦ ♣♦r exp(G)✱ ❝♦♠♦ s❡♥❞♦ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♥tr❡ ❛s ♦r❞❡♥s ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ◗✉❛♥❞♦ ❡ss❡ ❣r✉♣♦ é ✉♠ p✲❣r✉♣♦✱ ♦ ❡①♣♦❡♥t❡ s❡rá ❛ ♠❛✐♦r ♦r❞❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ G✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ G é ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✱ ❡♥tã♦ exp(G) = pn✱ ♣❛r❛ ❛❧❣✉♠ ♥❛t✉r❛❧ n✳
❉❡✜♥✐çã♦ ✷✳✶✳✺✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ P❛r❛ q✉❛❧q✉❡r i≥0 ❞❡✜♥✐♠♦s
Ωi(G) = hx∈G|xp
i = 1i
❡
Gpi =hxpi|x∈Gi.
❊ss❡s s✉❜❣r✉♣♦s sã♦ ❝❛r❛❝t❡ríst✐❝♦s ❡♠G✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❡①♣♦❡♥t❡✱ s❡ exp(G) = pn✱
❡♥tã♦ xpn
= 1♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦x❞❡G✳ ❆ss✐♠ t❡♠♦s q✉❡Ωn(G) = G✳ ❈♦♠ ✐ss♦✱ t❡♠♦s
❛ s❡❣✉✐♥t❡ sér✐❡ ❛s❝❡♥❞❡♥t❡✱ ❞❡♥♦♠✐♥❛❞❛ Ω✲sér✐❡ ❞❡ G
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✶✽
❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ Gpn
= hxpn
| x ∈ Gi = 1 ❡ ❛ss✐♠ t❡♠♦s ❛ sér✐❡ ❞❡s❝❡♥❞❡♥t❡✱ ❞❡♥♦♠✐♥❛❞❛ G✲sér✐❡
G=Gp0 >Gp1 >· · ·>Gpn−1
>Gpn = 1. ❈♦♠♦ ♦ s✉❜❣r✉♣♦ ❞❡ ❋r❛tt✐♥✐ é ❡str✐t♦ t❡♠♦s q✉❡ Gpi+1
6 (Gpi
)p 6 Φ(Gpi
) < Gpi ✳ ❈♦♠ ✐ss♦ ❛ G✲sér✐❡ ❞❡ ✉♠ p✲❣r✉♣♦ é ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡✳ ❊♥tã♦ ❛ G✲sér✐❡ ❞❡ ✉♠ p✲❣r✉♣♦ ❞❡ ❡①♣♦❡♥t❡ pn t❡♠ ❡①❛t❛♠❡♥t❡ n ♣❛ss♦s✳
❚❛♠❜é♠ ❞❡✜♥✐♠♦s ♦s s❡❣✉✐♥t❡s s✉❜❝♦♥❥✉♥t♦s✿
❉❡✜♥✐çã♦ ✷✳✶✳✻✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ P❛r❛ q✉❛❧q✉❡r i≥0 ❞❡✜♥✐♠♦s
Ω{i}(G) = {x∈G | xp
i = 1}
❡
G{pi} ={gpi | g ∈G}.
◗✉❛♥❞♦ ♦ ❣r✉♣♦ é ❛❜❡❧✐❛♥♦✱ ❝❧❛r❛♠❡♥t❡✱ ❡ss❡s s✉❜❝♦♥❥✉♥t♦s ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s s✉❜❣r✉✲ ♣♦s ❞❡✜♥✐❞♦s ❡♠ ✷✳✶✳✺✱ ♠❛s ♥ã♦ ❛♣❡♥❛s ♥❡❧❡s✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥♦s ♠♦str❛ ❛ ✐❣✉❛❧❞❛❞❡ ❞❡ss❡s s✉❜❣r✉♣♦s ❡ ✉♠❛ r❡❧❛çã♦ ❞♦ í♥❞✐❝❡ ❞❡ Ωi(G) ❡♠ G❡ ❛ ♦r❞❡♠ ❞❡ Gp
i
✱ ♦♥❞❡Gé ✉♠ p✲❣r✉♣♦ ❛❜❡❧✐❛♥♦✳
❚❡♥t❛r❡♠♦s ❞❡✐①❛r ❡①♣❧í❝✐t♦✱ ❡♠ ❝❛❞❛ ❝❛s♦✱ s❡ ❡st❛♠♦s tr❛t❛♥❞♦ ❞❡❧❡s ❝♦♠♦ ❣❡r❛❞♦ ♦✉ ❝♦♠♦ ❝♦♥❥✉♥t♦✳ ◆♦ ❝❛s♦ ❞❡ ❝♦✐♥❝✐❞✐r❡♠ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦✳
❚❡♦r❡♠❛ ✷✳✶✳✼ ✭❬✺❪✱ ❚❡♦r❡♠❛ ✷✳✸✮✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ P❛r❛ q✉❛❧q✉❡r i ≥ 0 t❡♠♦s q✉❡✿
(i) Ωi(G) ={x∈G | xp
i = 1}✳
(ii) Gpi
={xpi
| x∈G}✳
(iii) |G: Ωi(G)|=|Gp
i
| ✭❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ t❛♠❜é♠ |G:Gpi
|=|Ωi(G)|✮✳
❖❜s❡r✈❡ q✉❡ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ♥♦s ❞✐③ q✉❡ s❡Gé ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ ❛s três ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s✳ ▼❛s✱ s❡ ❡❧❛s sã♦ s❛t✐s❢❡✐t❛s ♥ã♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡Gé ❛❜❡❧✐❛♥♦✳ ◗✉❛♥❞♦ ✉♠ ❣r✉♣♦ s❛t✐s❢❛③ ❡ss❛s três ♣r♦♣r✐❡❞❛❞❡s✱ ♣❛r❛ t♦❞♦ i✱ ✉♠ ❞♦s ❛rt✐❣♦s tr❛❜❛❧❤❛❞♦s✱❬✽❪✱ ♦ ❞❡♥♦♠✐♥❛ ♣♦✇❡r ❛❜❡❧✐❛♥✳
❆♦ ❧♦♥❣♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ ✈❡r❡♠♦s ❝❛s♦s ♥♦s q✉❛✐s ♣❡❞✐r❡♠♦s ♦✉tr❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❛♦ ✐♥✈és ❞❡ ❛❜❡❧✐❛♥♦✱ ❞❡ ♠♦❞♦ q✉❡ ❡ss❡ t❡♦r❡♠❛ ❛✐♥❞❛ s❡❥❛ s❛t✐s❢❡✐t♦✳
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✶✾
❚❡♦r❡♠❛ ✷✳✶✳✽✳ ❙❡❥❛♠ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ❡ N✱ M s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ G✳ ❙❡ N 6
M[N, G]Np✱ ❡♥tã♦ N 6M✳
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ M ❡ N s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ G✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ M = 1✱ ❡♥tã♦ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ N = 1✳
❙✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡N 6= 1✳ ❈♦♠♦ ❡st❛♠♦s ❡♠ ✉♠p✲❣r✉♣♦✱ ❡①✐st❡ ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ K ❞❡ G t❛❧ q✉❡ |N : K| = p✳ ❚❡♠♦s q✉❡ K, N E G ❡ K < N✱ ❡♥tã♦ ❢❛③❡♥❞♦ q✉♦❝✐❡♥t❡ ♣♦r K✱ t❡♠♦s q✉❡ N/K EG/K✳
❆❣♦r❛✱ ❝♦♠♦ G/K é ✉♠ p✲❣r✉♣♦ ❡ |N : K| = p✱ ❡♥tã♦ N/K ∩Z(G/N) 6= 1✱ ❡ ✐ss♦ ❛❝❛rr❡t❛ q✉❡ N/K 6Z(G/N)✳ ❱♦❧t❛♥❞♦ ♣❛r❛ G✱ t❡♠♦s q✉❡ [N, G]6K✳
❚❛♠❜é♠ t❡♠♦s q✉❡Np 6K✳ ❉❡ ❢❛t♦✱ ❝♦♠♦|N/K|=p✱ ❡♥tã♦ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡N/K
t❡♠ ♦r❞❡♠ p ❡✱ ❛ss✐♠✱ (nK)p = npK = K✳ ▼❛s✱ npK = K s❡✱ ❡ s♦♠❡♥t❡ s❡✱ np ∈ K✳
▲♦❣♦✱ Np 6K✳
❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ Np 6K ❡ [N, G]6 K✱ ❡♥tã♦ Np[N, G] 6K✳ ▼❛s ✐ss♦ é ✉♠
❛❜s✉r❞♦✱ ♣♦✐s ♣♦r ❤✐♣ót❡s❡ N 6[N, G]Np ❡ K < N✳ P♦rt❛♥t♦✱ N = 1✳
◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ✈✐♠♦s ❛ ❋ór♠✉❧❛ ❞❡ ❈♦♠♣✐❧❛çã♦ ❞❡ ❍❛❧❧✳ ❆❣♦r❛ ✈❡r❡♠♦s ♠❛✐s ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ss❛ ❢ór♠✉❧❛ ✉t✐❧✐③❛❞❛s ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❆ ♣r✐♠❡✐r❛ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❈♦r♦❧ár✐♦ ✶✳✶✳✶✺ ❡ ❡❧❛ ♥♦s ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❞❡ ♣❡rt✐♥ê♥❝✐❛ ❡♥tr❡ ♦s t❡r♠♦s ❞❛ Ω✲sér✐❡ ❞♦ ❣r✉♣♦ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❞❛ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r ❛♣❧✐❝❛❞❛s ♥❡ss❡s t❡r♠♦s✳
❈♦r♦❧ár✐♦ ✷✳✶✳✾✳ ❙❡❥❛ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦✳ ❊♥tã♦ t❡♠♦s q✉❡
Ωpin(G)6Ωi−n(G)γ2(Ωi)p
n
γp(Ωi)p
n−1
γp2(Ωi)p
n−2
γp3(Ωi)p
n−3
. . . γpn(Ωi) ♦♥❞❡ Ωl = Ωl(G) ♣❛r❛ l≤1 ❡ Ωl = 1 ♣❛r❛ l ≤0✳
▲❡♠❜r❡✲s❡ q✉❡ ❞❡✜♥✐♠♦s s✉❜❣r✉♣♦ ❝♦♠✉t❛❞♦r ❞❡ q✉❛❧q✉❡r ❝♦♠♣r✐♠❡♥t♦✳ ❈♦♥s✐❞❡✲ r❛♥❞♦ N ❡H s✉❜❣r✉♣♦s ❞❡G✈❛♠♦s ❛❞♦t❛r ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦[H,kN]♣❛r❛ ♦ ❝♦♠✉t❛❞♦r
[H, N, . . . , N]✱ ♦♥❞❡ N ❛♣❛r❡❝❡k ✈❡③❡s✳
❆ ♦✉tr❛ ❝♦♥s❡q✉ê♥❝✐❛ q✉❡ ♦❝♦rr❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❡♠ t♦r♥♦ ❞❡ss❛ ❢ór♠✉❧❛✱ ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ é ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳
❚❡♦r❡♠❛ ✷✳✶✳✶✵✳ ❙❡❥❛♠ G ✉♠ p✲❣r✉♣♦ ✜♥✐t♦ ❡ N✱ M s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❞❡ ●✳ ❊♥tã♦✱ ♣❛r❛ ✉♠ ♥❛t✉r❛❧ k✱
[Npk, M]≡[N, M]pk( ♠♦❞ [M,pN]p
k−1
[M,p2N]p
k−2
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✷✵
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ n∈N ❡m ∈M✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✳✶✹✱ t❡♠♦s
[npk, m]≡[n, m]pk( ♠♦❞ γ2(L)p
k γp(L)p
k−1
γp2(L)p
k−2
. . . γpk(L)),
♦♥❞❡ L=hn,[n, m]i✳ ❖❜s❡r✈❡ q✉❡γ2(L) = h[l1, l2] | l1, l2 ∈Li6[M, N, N]✳ ❉❡ss❛ ❢♦r♠❛✱
t❡♠♦s ❛ s❡❣✉✐♥t❡ ✐♥❝❧✉sã♦
γ2(L)p
k
6([M, N, N])pk 6[M, N]pk ❡✱ ❛ss✐♠✱ γpi(L)p
k−i
= [γ2(L),pi−2(L)]p k−i
6 [M, N, N,pi−2N]p k−i
= [M,piN]p k−i
✱ ♣❛r❛ ♦s ❞❡♠❛✐s t❡r♠♦s✱ ❝♦♠ i= 1, . . . , k✳
❈♦♠ ✐ss♦✱ ♣♦r ✉♠ ❧❛❞♦✱ t❡♠♦s q✉❡
[npk, m]∈[n, m]pk[N, M]pk
k
Y
i=1
[M,piN]p k−i
.
❚♦♠❛♥❞♦ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ M ❡N s❡❣✉❡ q✉❡
[Npk, M]6[N, M]pk
k
Y
i=1
[M,piN]p k−i
.
P♦r ♦✉tr♦ ❧❛❞♦✱
[N, M]pk 6[Npk, M][M, N, N]pk
k
Y
i=1
[M,piN]p k−i
.
❱❛♠♦s ♠♦str❛r ❛ ♦✉tr❛ ✐♥❝❧✉sã♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ❛ ♦r❞❡♠ ❞❡N✳ ❙❡|N|= 1✱ ❡♥tã♦ ❛ ✐♥❝❧✉sã♦ é ❝❧❛r❛♠❡♥t❡ ✈á❧✐❞❛✳ P❛r❛ ♦ ♣❛ss♦ ❞❡ ✐♥❞✉çã♦✱ ❧❡♠❜r❡ q✉❡ [N, M]6[N, G]< N✱ ❡♥tã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❡♠ [M, N]❡ ✐ss♦ ❛❝❛rr❡t❛
[[M, N], N]pk 6[[M, N]pk, N]
k
Y
i=1
[N,pi[M, N]]p k−i
6[[M, N]pk, N]
k
Y
i=1
[M,piN]p k−i
.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱
[N, M]pk 6[Npk, M][M, N, N]pk
k
Y
i=1
[M,piN]p k−i
6[Npk, M][[M, N]pk, N]
k
Y
i=1
[M,piN]p k−i
[N, M]pk 6[Npk, M][[N, M]pk, G]
k
Y
i=1
[M,piN]p k−i
✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✷✶
P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✳✽✱ ✈❛❧❡ ❛ ✐♥❝❧✉sã♦ ❞❡s❡❥❛❞❛✳ P♦rt❛♥t♦✱ ✈❛❧❡ ❛ ❝♦♥❣r✉ê♥❝✐❛
[N, M]pk ≡[Npk, M](♠♦❞ [M,pN]p
k−1
[M,p2N]p
k−2
