C´esarPolcinoMilies SymmetricandSkew-SymmetricElementsinGroupRings

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Notation and Basic Facts Commutativity of Symmetric Elements Commutativity of Skew-symmetric Elements Anticommutativity Lie Properties ofKG Nilpotence of skew symmetrics

Symmetric and Skew-Symmetric Elements in Group Rings

C´esar Polcino Milies

Universidade de S˜ao Paulo

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Notation

LerR be a ring. A map^∗ :R →R is aninvolution if (i) (α+β)^∗=α^∗+β^∗,

(ii) (αβ)^∗=β^∗α^∗, and (iii) α^∗∗=α.

R⁺={r ∈R |r^∗ =r} is symmetric elementsof R under ^∗. R⁻={r ∈R |r^∗ =−r} skew symmetric elements.

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Notation

(ii) (αβ)^∗=β^∗α^∗, and (iii) α^∗∗=α.

R⁺={r ∈R |r^∗ =r} is symmetric elementsof R under ^∗.

R⁻={r ∈R |r^∗ =−r} skew symmetric elements.

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Notation

(ii) (αβ)^∗=β^∗α^∗, and (iii) α^∗∗=α.

R⁺={r ∈R |r^∗ =r} is symmetric elementsof R under ^∗. R⁻={r ∈R |r^∗ =−r} skew symmetric elements.

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Setting

R will denote a commutative ring with unity.

G a group.

RG the group ring of G overR.

Example: The Classical Involutionis defined extending linearly the mapg 7→g⁻¹ to RG:

α= X

g∈G

αgg 7→ α^∗ =X

g∈G

αgg⁻¹.

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Setting

G a group.

α= X

g∈G

αgg 7→ α^∗ =X

g∈G

αgg⁻¹.

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Setting

G a group.

α= X

g∈G

αgg 7→ α^∗ =X

g∈G

αgg⁻¹.

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Setting

G a group.

α= X

g∈G

αgg 7→ α^∗ =X

g∈G

αgg⁻¹.

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Setting

G a group.

α= X

g∈G

α_gg 7→ α^∗ =X

g∈G

α_gg⁻¹.

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Commutativity of Symmetric Elements

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For any ringR, with an involution^∗, ifα, β ∈R⁺ then

αβ∈R⁺ ⇐⇒ αβ =βα.

V. Bovdi, L. G. Kov´acs and S. K. Sehgal, in 1996, gave conditions for the group of symmetric units (under the classical involution) of a modular group ringRG to be a subgroup, assuming thatG is a locally finitep-group andR a commutative ring of prime

characteristicp.

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For any ringR, with an involution^∗, ifα, β ∈R⁺ then

αβ∈R⁺ ⇐⇒ αβ =βα.

V. Bovdi, L. G. Kov´acs and S. K. Sehgal, in 1996, gave conditions for the group of symmetric units (under the classical involution) of a modular group ringRG to be a subgroup, assuming thatG is a locally finitep-group andR a commutative ring of prime

characteristicp.

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Notice that an elementα of RG is symmetric if and only if α^∗ =



 X

g∈G

α(g)g





∗

=X

g∈G

α(g)g⁻¹= X

g∈G

α(g)g =α;

i.e., if and only ifα(g) =α(g⁻¹), for allg ∈G.

Thus,RG⁺ is generated, as an R−module, by the set {g +g⁻¹ |g ∈G,g²6= 1} ∪ {g ∈G |g² = 1}.

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

 X

g∈G

α(g)g





∗

=X

g∈G

α(g)g⁻¹= X

g∈G

α(g)g =α;

i.e., if and only ifα(g) =α(g⁻¹), for allg ∈G. Thus,RG⁺ is generated, as an R−module, by the set

{g +g⁻¹ |g ∈G,g²6= 1} ∪ {g ∈G |g² = 1}.

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

 X

g∈G

α(g)g





∗

=X

g∈G

α(g)g⁻¹= X

g∈G

α(g)g =α;

i.e., if and only ifα(g) =α(g⁻¹), for allg ∈G. Thus,RG⁺ is generated, as an R−module, by the set

{g +g⁻¹ |g ∈G,g²6= 1} ∪ {g ∈G |g² = 1}.

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Theorem (Broche, 2006)

LetG be a group and letR be a commutative ring with unity of characteristic different from 2. Then, the following are equivalent:

1 The symmetric elements ofRG commute.

2 G is either an abelian group or a Hamiltonian 2-group.

3 The symmetric units of RG commute.

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Theorem (Broche, 2006)

LetG be a group and letR be a commutative ring with unity of characteristic equal to 2. Then, the symmetric elements ofRG commute if and only ifG is either an abelian group or the direct product of an elementary abelian 2-group and a groupH, for which one of the following conditions holds.

(i) H has an abelian subgroup Aof index 2 and an elementb of order 4, such that the conjugation forb inverts each element of A;

(ii) H is the direct product of the Quaternion group of order 8 and the cyclic group of order 4, or the direct product of two Quaternion groups of order 8;

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(iii) H is the central product of the group

hx,y |x⁴ =y⁴ = 1, x² = (y,x)iwith the quatenion group of order 8, where the nontrivial element common to the two central factors isx²y²;

(iv) H is isomorphic to one of the groupsH₃₂ andH₂₄₅, where

H₃₂=hx,y,u |x⁴ =y⁴ = 1,x²= (y,x), y² =u² = (u,x),x²y² = (u,y)i, and

H245=hx,y,u,v |x⁴ =y⁴ = (v,u) = 1, x² =v² = (y,x) = (v,y),

y² =u² = (u,x),x²y² = (u,y) = (v,x)i.

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Ifϕ:G →G is any involution ofG, then the map

α= X

g∈G

α_gg 7→ α^∗ =X

g∈G

α_gϕ(g) is an involution ofRG.

We shall denote byG⁺ andRG⁺ the set of symmetric elements of G andRG respectively.

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α= X

g∈G

α_gg 7→ α^∗ =X

g∈G

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α= X

g∈G

α_gg 7→ α^∗ =X

g∈G

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Theorem (Jespers and Ruiz, 2006)

Letϕbe an involution of a non-abelian groupG and let R be a commutative ring os characteristic different from 2. Then, the following are equivalent:

1 RG⁺ is commutative.

2 The group G has the LC property, a unique nontrivial commutator s and the involutionϕis given by:

ϕ(g) =

g ifg ∈ Z(G) sg ifg 6∈ Z(G).

3 G/Z(G)∼=C2×C2,ϕ(g) =g ifg ∈ Z(G) and ϕ(g) =h⁻¹gh,for anyh not commuting with g.

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We recall that, given a groupG, an integral domain R is called G-favoravelif for every element g ∈G of finite order, there exists an elementα∈R such that 1−α^o(g⁾ is invertible inR. Notice that every infinite field is G-favoravel.

Theorem (Jespers and Ruiz 2006)

LetG be a periodic group and R a G-favoravel integral domain. Then the following conditions are equivalent:

1 U(RG)⁺ is a subgroup.

2 RG⁺ is a subring.

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We recall that, given a groupG, an integral domain R is called G-favoravelif for every element g ∈G of finite order, there exists an elementα∈R such that 1−α^o(g⁾ is invertible inR. Notice that every infinite field is G-favoravel.

Theorem (Jespers and Ruiz 2006)

LetG be a periodic group and R a G-favoravel integral domain.

Then the following conditions are equivalent:

1 U(RG)⁺ is a subgroup.

2 RG⁺ is a subring.

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Commutativity of Skew-symmetric Elements

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Why study commutativity of skew-symmetric elements?

We denote by

U_∗ ={u ∈ U(A)|uu^∗ = 1} theunitary unitsofRG.

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We denote by

U_∗ ={u ∈ U(A)|uu^∗ = 1}

theunitary unitsofRG.

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We denote by

U_∗ ={u ∈ U(A)|uu^∗ = 1}

theunitary unitsofRG.

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Any associative algebraA, can be regarded as a Lie algebra with the same addition and the product defined by:

[α, β] =αβ−βα, for all α, β ∈A.

Notice thatA⁻ is a Lie subalgebraof A,

For elementsα₁, α₂, . . . , αn−1, α_n∈A we define, inductively, [α1, α2, . . . αn−1, αn] = [[α1, α2, . . . , αn−1], αn]. DefinitionWe say thatA isLie Nilpotent if there exists a positive integern such that [α1, α2, . . . , αn] = 0, for all α₁, α₂, . . . , α_n∈KG

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For elementsα₁, α₂, . . . , αn−1, α_n∈A we define, inductively, [α1, α2, . . . αn−1, αn] = [[α1, α2, . . . , αn−1], αn]. DefinitionWe say thatA isLie Nilpotent if there exists a positive integern such that [α1, α2, . . . , αn] = 0, for all α₁, α₂, . . . , α_n∈KG

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For elementsα₁, α₂, . . . , αn−1, α_n∈A we define, inductively, [α1, α2, . . . αn−1, αn] = [[α1, α2, . . . , αn−1], αn].

DefinitionWe say thatA isLie Nilpotent if there exists a positive integern such that [α1, α2, . . . , αn] = 0, for all α₁, α₂, . . . , α_n∈KG

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For elementsα₁, α₂, . . . , αn−1, α_n∈A we define, inductively, [α1, α2, . . . αn−1, αn] = [[α1, α2, . . . , αn−1], αn].

DefinitionWe say thatA isLie Nilpotent if there exists a positive integern such that [α1, α2, . . . , αn] = 0, for all α₁, α₂, . . . , α_n∈KG

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Theorem (Giambruno and P.M. 2003)

LetAbe a finite dimensional algebra over an algebraically closed fieldF which is nonabsolute and char F 6= 2. Then, the following are equivalent.

(i) A⁻ is commutative.

(ii) A⁻ is Lie nilpotent.

(iii) U_∗(A) does not contain free groups of rank 2.

(iv) U_∗(A) satisfies a group identity.

(v) There exists a positive integer m such that (x₁²,x₂²)^m ≡1 is a group identity for U_∗(A).

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Theorem (O. Broche, A. Dooms and M. Ruiz 2007) Ais a semisimple algebra over a non-absolute fieldK with char(K)6= 2 then

U∗(A) satisfies a GI ⇔(A)⁻ is commutative⇔ U∗(A) is abelian

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Notation: For a given ringR, we denoteR2={r ∈R |2r = 0}.

Notice that as anR-module, (RG)⁻ is generated by the set S = {g−ϕ(g)| g ∈G \G⁺} ∪ {rg | g ∈G⁺, r ∈R2} Therefore (RG)⁻ is commutative if and only if the elements inS commute.

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Notation: For a given ringR, we denoteR2={r ∈R |2r = 0}.

Notice that as anR-module, (RG)⁻ is generated by the set S = {g−ϕ(g)| g ∈G \G⁺} ∪ {rg | g ∈G⁺, r ∈R2} Therefore (RG)⁻ is commutative if and only if the elements inS commute.

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Theorem (Broche, Jespers, P.M. and Ruiz 2009)

LetR be a commutative ring. SupposeG is a non-abelian group andϕis an involution on G. Then, (RG)⁻ is commutative if and only if one of the following conditions holds:

1 K =hg ∈G|g 6∈G⁺i is abelian (and thusG =K ∪Kx, wherex ∈G⁺, andϕ(k) =xkx⁻¹ for all k ∈K) and R₂² ={0}.

2 R2={0} andG contains an abelian subgroup of index 2 that is contained in G⁺.

3 char(R) = 4,|G⁰|= 2, G/G⁰ = (G/G⁰)⁺,g² ∈G⁺ for all g ∈G, andG⁺ is commutative in case R₂²6={0}.

4 char(R) = 3,|G⁰|= 3, G/G⁰ = (G/G⁰)⁺ and g³∈G⁺ for all g ∈G.

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Anticommutativity of Symmetric and Skew-symmetric Elements

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For any ringR, with an involution^∗, ifα, β ∈R⁻ then

αβ∈R⁻ ⇐⇒ αβ=−βα.

Hence,R⁻ is a subring if and only if skew-symmetric elements anticommute.

Lemma (Goodaire and P.M.)

Letg 7→g^∗ denote an involution of a groupG and let∗ also denote the linear extension toRG. If the elements of RG that are skew-symmetric relative to∗anticommute, then one of the following holds:

1 the characteristic ofR is 2,

2 the characteristic of R is 4 and gg^∗ =g^∗g and (g^∗)² =g² for allg ∈G,

3 G is abelian and∗ is the identity onG.

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2 the characteristic of R is 4 and gg^∗ =g^∗g and (g^∗)² =g² for allg ∈G,

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2 the characteristic of R is 4 and gg^∗=g^∗g and (g^∗)² =g² for allg ∈G,

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Theorem (Goodaire and P.M.)

LetG be a group with an involution ∗and let R be a coefficient ring of characteristic 4. Then the set (RG)⁻ of elements that are skew-symmetric with respect to∗ forms a subring ofRG if and only if

1 G is an abelian group and there existss ∈G with s² = 1 and g^∗ =g orsg for all g ∈G (this includes the possibility that ∗ is the identity onG);

or

2 G is a nonabelian group with a unique nonidentity

commutator, s (necessarily central and of order 2), for every g ∈G,g^∗=g orsg, and either symmetric elements commute or whenever α, β∈R satisfy 2α= 2β= 0, thenαβ = 0.

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When do symmetric elements anticommute?

Lema

Letg 7→g^∗ denote an involution of a groupG and let∗ also denote the linear extension toRG. If the elements of RG that are symmetric relative to∗ anticommute, then

1 charR = 2, or

2 charR = 4 and gg^∗=g^∗g andg² = (g^∗)² for allg ∈G, or

3 charR = 8, G is abelian, andg^∗ =g for allg ∈G.

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When do symmetric elements anticommute?

Lema

Letg 7→g^∗ denote an involution of a groupG and let∗ also denote the linear extension toRG. If the elements of RG that are symmetric relative to∗ anticommute, then

1 charR = 2, or

2 charR = 4 and gg^∗=g^∗g andg² = (g^∗)² for allg ∈G, or

3 charR = 8, G is abelian, andg^∗ =g for allg ∈G.

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Theorem (Goodaire and P.M.)

LetG be a group with an involution ∗and let R be a coefficient ring of characteristic 4. Then the set (RG)⁺ of symmetric elements is anticommutative if and only if

1 G is abelian and there existss ∈G with s² = 1 andg^∗ =g or sg for all g ∈G;

or

2 G is a nonabelian group with a unique commutators 6= 1 and g^∗ =g org^∗ =sg for all g ∈G.

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Lie Properties of KG

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Theorem (Passi, Passmann and Sehgal, 1973)

LetK be a field of characteristic p. Then KG is Lie nilpotent if and only if G is nilpotent and p-abelian;

Theorem (Sehgal, 1978)

LetK be a field of characteristic p6= 2. Then KG is Lie n-Engel if and only if

1)p= 0 andG is abelian.

2)p>0, G is nilpotent and contains a normal p-abelian subgroup Asuch thatG/A is a finite p-group.

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Theorem (Passi, Passmann and Sehgal, 1973)

LetK be a field of characteristic p. Then KG is Lie nilpotent if and only if G is nilpotent and p-abelian;

Theorem (Sehgal, 1978)

LetK be a field of characteristic p6= 2. Then KG is Lie n-Engel if and only if

1)p= 0 andG is abelian.

2)p>0, G is nilpotent and contains a normal p-abelian subgroup Asuch thatG/A is a finite p-group.

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Lie Nilpotence of skew symmetrics

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Theoren Giambruno and Sehgal (1993)

LetR be a seimiprime ring with involution such that 2R =R. If the Lie subalgebraR⁻ is Lie nilpotent, then it is commutative; i.e.

[R⁻,R⁻] = 0 (and, in this case,R satisfies the standard pollynomial identity of degree 4).

Corollary

Assume that FG is semiprime, thatchar(F)6= 2 and that G is a torsion group with no elements of order 2, Then FG is Lie nilpotent if and only if G is abelian.

Notice that, ifchar(F) = 0 thenFG is semiprime, Hence, from now onwe shall assume that char(F)>2

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Corollary

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Corollary

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Tools

Letf(x1,x₁^∗,x2,x₂^∗, . . . ,xn,x_n^∗) be a non-trivial polynomial in the free associative algebra with involutionK{X,∗}. We recall that f is a *-polynomial identity for an algebra with involution Aif f(a1,a^∗₁,a2,a^∗₂, . . . ,an,a^∗_n) = 0 for all a1, . . .an in A.

Theorem (Amitsur 1968)

If an algebraA with involution satisfies a *-polynomial identity, then it satisfies a polynomial identity.

Theorem (Isaacs-Passman (1964) and Passman (1972)) LetK be a fiald of characteristicp ≥0. Then KG satisfies a polynomial identity if and only ifG contains ap-abelian subgroup Aof finite index

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Tools

Letf(x1,x₁^∗,x2,x₂^∗, . . . ,xn,x_n^∗) be a non-trivial polynomial in the free associative algebra with involutionK{X,∗}. We recall that f is a*-polynomial identity for an algebra with involutionA if f(a1,a^∗₁,a2,a^∗₂, . . . ,an,a^∗_n) = 0 for all a1, . . .an in A.

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Tools

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Tools

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Theorem (Giambruno, P.M. and Sehgal, 2009)

LetG be a group with no elements of order 2 and K a field of characteristicp 6= 2. Then,

(KG)⁺ is Lien-Engel⇐⇒KG is Lie m-Engel, for somem>0.

(KG)⁺ is Lie nilpotent ⇐⇒KG is Lie nilpotent.

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Tools

G torsion, [G :A] finite andA p-abelian implyG locally finite.

Aof finite index⇒A∩A^∗ of finite index.

Hence, we can always assume thatG contains ap-abelian subgroup of finite indexthat is invariant under^∗.

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Tools

G torsion, [G :A] finite andA p-abelian implyG locally finite.

Aof finite index⇒A∩A^∗ of finite index.

Hence, we can always assume thatG contains ap-abelian subgroup of finite indexthat is invariant under^∗.

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Tools

Assume thatAis an abelian group with no elements of order 2 and let^∗ :A→A be an automorphism of order 2. Then

A²=A₁×A₂; where

A₁ ={a∈A|a^∗ =a}

and

A2 ={a∈A|a^∗ =a⁻¹}.

Moreover ifAis a torsion group, A=A1×A2.

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The finite case

Theorem

LetG be a finite group of odd order. ThenFG⁻ is Lie nilpotent if and only if eitherFG is Lie nilpotent orcharF =p >2,P is a subgroup,G/P is abelian and^∗ is trivial on G/P.

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Theorem

LetF be a field of characteristic p>2 and G a torsion group with no elements of order 2: Let^∗ be an involution on FG induced by an involution ofG. Then the Lie algebra FG⁻ is nilpotent if and only ifFG is Lie nilpotent orcharF =p >2 and the following conditions hold:

(i) The setP of p-elements inG is a subgroup, (ii) ^∗ is trivial on G/P,

(iii) there exist normal ^∗-invariant subgroupsAandB,B ⊂Asuch that B is a finite central p-subgroup ofG,A/B is central in G/B with both G/Aand{a∈A|aa^∗ ∈B}finite.

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C´esarPolcinoMilies SymmetricandSkew-SymmetricElementsinGroupRings

Symmetric and Skew-Symmetric Elements in Group Rings

Tools

Tools

Tools

Tools

Tools

Tools

Tools

The finite case

Thank You!!