On this webpage, users can access a development archive hosted on GitHub, contribute to the improvement of the software and ask questions about its functionality. This experience gives us a significant amount of insight to write a comprehensive and user-friendly presentation of the capabilities of this computer algebra system.
The very first steps
We will also discuss some important algebraic structures, such as the integers and rational numbers, finite fields, sets and permutations, matrices, polynomials and vector spaces. In the UNIX-based operating system, to run aGAPscript from the terminal, we simply write
Basic arithmetic
Basic programming
- Objects and variables
- Conditionals
- Functions
- Loops
- Strings
- Lists
- Ranges
- Sets
- Records
To remove elements from a list use Remove: gap> # Remove the first element from the list gap> Remove(primes, 1);;. Similarly, ForAll returns true if all elements of the list meet the required condition and false otherwise.
Other numbers
Floating–point numbers
The function NanosecondsSinceEpoche returns the time (in nanoseconds) that has elapsed since a fixed (and unknown) time in the past. In the following code, we calculate the number of milliseconds we need to create a large random matrix and calculate its determinant:
Finite fields
If𝑝 is a prime number, it is natural to identify the field F𝑝with the ringZ/𝑝of integers modulo𝑝.
Cyclotomic numbers
Permutations
Conversely, any list representing a permutation (i.e., of size 𝑛containing the integers from 1to𝑛) can be converted into a permutation with the functionPermList.
Matrices
This function returns if the matrix is not invertible. IsIdentityMat returns either true if the argument is the identity matrix or false otherwise. The trace of a square matrix is computed with Trace, the determinant with Determinant, and the rank with Rank.
Polynomials
The function NullspaceMat computes the vector space generated by the solutions of𝑥 𝐴=0, where𝑥 is a matrix of size 1×𝑚and𝐴 is a matrix of size𝑚×𝑛. The degree of a polynomial can be obtained with Degree, and the coefficients with the function CoefficientsOf Univariate Polynomial.
Vector spaces
Problems
For example, if the argument is list[1,4,2], the function should appear. 1.17.Write a function that, given a list of words, returns the longest one. 1.18.Write a function that returns the average value of a given list of numbers. 1.19.Write a function that, given a letter, returns whether the letter is a vowel and vice versa. After you guess how each term is calculated, write a script to generate the first terms of the sequence. as a product of disconnected cycles.
Basic constructions
More generally, we can construct any finite abelian group by specifying the orders of the cyclic factors. Example 2.22. Let's see that S3 contains three conjugation classes with representative side, (1 2) and (1 2 3), so that. A naive idea to prove that A4 has no subgroups of order six is to consider all the.
Here we have another idea: if A4 has a subgroup of order six, then the index of this subgroup in A4 is two. Now let's use the commutator to prove that A4 has no subgroups of order six. If there exists a subgroup𝐾 of order six, then 𝐾 is normal in A4 and the quotient A4/𝐾 is cyclic of order two.
Group actions
In this case we get two orbits and not three, since the permutation representation ignores fixed points.
Homomorphisms
Example 2.39. Let's construct the cyclic group𝐶12with generator𝑔as a group of permutations, the subgroup𝐾 =h𝑔6and the quotient𝐶12/𝐾. Example 2.40. With GQuotients we can determine when a given group is an epimorphic image of another one. So far, there is no known proof of the Schreier conjecture that is independent of the classification of simple groups.
Semidirect products
Example 2.58. Let us see how the alternating group A5 acts on a convolved space by right multiplication.
Solvable groups
This happens because SubgroupsSolvableGroupe has created a list of subgroups of 𝑃 that contains all subgroups of order25 and (perhaps) other subgroups of 𝑃. This does not mean that it is always more convenient to use this function when it comes to solvable groups. Note that SubgroupsSolvableGroup again produced a list of subgroups containing all normal subgroups of 𝑃 and (perhaps) some other non-normal subgroups of 𝑃.
Hall's theorem states that if 𝐺 is a finite solvable group of order 𝑎 𝑏 with gcd(𝑎, 𝑏)=1, then there exists a unique (nonempty) conjugate class of subgroups of order𝑎. Such order subgroups are called Hall𝜋-subgroups of𝐺, where𝜋 is the set of prime divisors of𝑎. The fact that these groups are not isomorphic also follows from the structure of Hall's subgroups:
Finitely presented groups
Some of the features we used earlier can also be used in free groups. Example 2.66. For positive integers𝑙 , 𝑚, 𝑛, we define a von Dyck group (or triangular group) of type (𝑙 , 𝑚, 𝑛) as a group. Novikov proved that there exists a finitely represented group such that the word problem for this group is undecidable (see [44]).
Burnside's problem can be stated as follows: For what positive integer 𝑛is the free Burnside group𝐵(2, 𝑛)finite. Note that our groupB23 is exactly the group 𝐵(2,3), since every non-trivial element has order three. We will combine computer calculations with different theorems to recover the structure of the center, the commutator subgroup and the respective quotients.
Problems
It is known that the probability that two elements of 𝐺 commute is prob(𝐺) = 𝑘/|𝐺|. We begin with a brief introduction to the group databases in the system, with a particular emphasis on the SmallGroup library. For example, we examine the celebrated McKay conjecture, the Thompson conjecture on products of conjugation classes, and a Harada conjecture in 2018 about which little is known.
At the end of the chapter, we give a brief introduction to group rings and give examples, including the computation of the Wedderburn decompositions with the package Wedderga. In that regard, we study a matrix representation proposed in Passman's book and present a script that checks Gardam's proof of the counterexample to the Kaplansky unity conjecture (namely, there are non-trivial units in the Promislow group grouping over the field of two elements).
Group databases
Example 3.3. In one rule we can check whether there are no simple groups of order 84. In Example 3.8, we could have tried to find a faithful representation of small degree using the function LesserDegreePermutationRepresentation. In [2] Berkovich proved that if 𝑝 > 2 is prime and 𝐺 is a finite𝑝 group of order𝑝𝑛and exponent𝑝, then.
Example 3.13.We prove that the smallest finite perfect group that is not quasisimple is a group of the form𝐶4. Example 3.15. Let us prove that there is no sharply 4-transitive group of degree seven or nine. Example 3.17 (Baer trick). Let 𝐺 be a finite nilpotent group of odd order and nilpotency class at most two.
Representations
We are working with 3×3 matrices, so it is better to use the function Display: gap> Display(Image(rho. Using maximal subgroups, we will easily describe the structure of the Sylow 5 subgroups: they are semidirect products of the form𝐶2. To achieve this goal, we first find the generators of the 10th maximal subgroup ofTh which has order 12000 and thus contains a Sylow 5 subgroup ofTh.
Can you try to calculate the structure of the Sylow 5 subgroups by doing the calculations directly inTh. Example 3.26 (Dade's solution to Brauer's problem).Dade's paper shows the existence of Brauer pairs of groups of order𝑝7, for𝑝 ≥5 a prime number. Skrzipczyk found pairs of non-isomorphic groups of order28 with the same character table and power map.
Some conjectures
- McKay’s conjecture
- Isaacs–Navarro conjecture
- Ore’s conjecture
- Thompson’s conjecture
- Szep’s conjecture
- Arad–Herzog conjecture
- Hughes’ conjecture
- Harada’s conjecture
- Berkovich’s conjecture
- Wall’s conjecture
- Quillen’s conjecture
With this function, we can easily check the guess on some small examples, such as SL2(3). Wilson proved that McKay's conjecture is true for simple sporadic groups using a description of the normalizers of their Sylow subgroups. However, when the function does not return true, it cannot be concluded that the conjecture is false.
While the original statement of the conjecture was proved for solvable groups by Quillen, the version mentioned in Conjecture 3.11 is still open even for solvable groups.
Group rings
Example 3.43. Let 𝐾 be a field of two elements and 𝐺 =h𝑔i a third-order cyclic group with generator 𝑔 = (1 2 3). The Jacobson radical 𝐽(𝑅) of 𝑅 is defined as the intersection of all left maximal ideals of 𝑅. Example 3.44. Let 𝐾 be a field of two elements, 𝐺=D8 a body-hedral group of the eighth order and 𝐴=𝐾[𝐺].
Example 3.46. Let𝐾 be the field of two elements and 𝐺 be the cyclic group of order two. Plesken proposed the study of a certain Lie subalgebra of the group ring with the classical Lie bracket; see [10]. Example 3.49. The Plesken Lie algebra of the alternating group A4 over the field of two elements has basis.
Kaplansky’s unit conjecture
Example 3.51. Now we check whether the Plesken Lie algebra 𝑄8 is semisimple and isomorphic to 𝔰𝔩2:. I LAGUNA package: Constructing a Lie algebra .. lt;a Lie algebra over rationals, with 8 generators>. lt;Lie algebra of dimension 0 over Rationals>. In this section, we work with the Promislow 𝑃 group and show that it does not satisfy the unique product property.
In the following example, we show that the Promislow group𝑃 does not have the unique product property; see [46]. does not have the unique product characteristic. In particular, this implies that each element𝛼 of𝐾[𝑃] can be written uniquely as 𝛼0+𝛼1𝑎+𝛼2𝑏+𝛼3𝑎 𝑏, where𝛼0,𝛼1,𝛼2,𝛼2,𝑦±1,𝑧±1]. Therefore, an element 𝛼=𝛼0+𝛼1𝑎+𝛼2𝑏+𝛼3𝑎 𝑏 is a trivial unit if and only if there exists an index𝑖such that𝛼𝑗 =0for 𝑗≝ , 𝑦±1, 𝑧±1].
Problems
The set of perfect subgroups of 𝐺. c) The set of quasi-simple subgroups of 𝐺. d) The layer of𝐺, i.e. the subgroup generated by the subnormal quasi-simple subgroups of𝐺. e) The generalized Fitting subgroup of 𝐺, that is, the subgroup generated by the Fitting subgroup and the layer of 𝐺. Now we present a second solution that does not use the structure of the normal subgroups of Aut(𝐿) and relies only on elementary facts of group theory. Next, we calculate the list 𝑋 of subgroups from 𝑃 to 𝑃 conjugation using the Subgroups SolvableGroup function.
To obtain all 𝑝 subgroups of 𝐺, we calculate the 𝐺 conjugation classes of the elements of 𝑋. The influence of the size of conjugation classes on the structure of finite groups: a study. Asian-Eur. Translated from the French original by John Stillwell, corrected 2nd edition of the 1980 English translation.
