Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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2 votes
2 answers
245 views

In which books we can find structure information for finite simple groups and their Schur covering groups?

In which books we can find representations or character tables, Sylow 2-subgroups and conjugacy classes for finite simple groups and their Schur covering groups and properties for Schur multiplier of ...
14 votes
2 answers
5k views

Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution, $\require{AMScd}$ \begin{CD} \cdots @>N>> \mathbb ...
2 votes
0 answers
94 views

Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
39 votes
7 answers
9k views

Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
2 votes
1 answer
159 views

Examples of 3-transitive expander family of Schreier graphs

What are examples of expander family of 3-transitive Schreier graphs? Meaning for an action that is 3-transitive. It is better to have an option for randomization. We know that choosing 2 elements ...
2 votes
0 answers
64 views

Is there a method to find the order of a lift of an element of order 2 to the Schur cover?

Let $G$ be a finite non-abelian simple group, $M.G$ the Schur covering group of $G$. Is there a method to find the order of a preimage of an element of order 2 in the natural homomorphism $\pi: M.G\...
7 votes
0 answers
172 views

Finite groups such that non-central, commuting elements have the same stabilizer

Let me say that a finite, non-abelian group $G$ has the property $(P)$ if for any two elements $x, \, y \in G - Z(G)$ such that $y \in C_G(x)$, one has $C_G(y)=C_G(x)$. For instance, it is easy ...
5 votes
2 answers
555 views

Orbits of independent sets of the hypercube

How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes? The counting of the number of independent sets in an $n$-dimensional ...
1 vote
1 answer
77 views

What do conjugacy classes of involutions like in finite simple group $E_7(q)$?

Are there any refences for conjugacy classes of involutions in finite simple group $E_7(q)$?
3 votes
1 answer
51 views

describing embedding $U_3(q)<O_6^-(q)$, $q$ even

Let $q=2^k$. I need to explicitly construct $U_3(q)$ as a subgroup of $G=GO_6^-(q)$. It is well-known that $G\cong U_4(q)$, and as a subgroup of the latter one has $U_3(q)$ fixing a non-isotropic ...
3 votes
0 answers
708 views

Differences between GAP and MAGMA [closed]

GAP and MAGMA are computer algebra systems. What are the objective differences between the two? Which capabilities are not shared? How do they compare on facilities for working with character tables?...
5 votes
1 answer
139 views

How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable

Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$. If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
5 votes
2 answers
1k views

Cardinality of certain subsets in vector spaces over finite fields

Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
7 votes
2 answers
3k views

Order of product of group elements

Let $G$ be a finite non-commutative group of order $N$, and let $x, y \in G$. Let $a$ and $b$ be the orders of $x$ and $y$, respectively. Can we say anything non-trivial about the order of $xy$ in ...
2 votes
0 answers
157 views

On automorphism group

Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $U_{n}$ denote the unitriangular group of $n\times n$ upper triangular matrices with ones on the diagonal, over $...
7 votes
1 answer
139 views

How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?

I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason? https://oeis.org/A002487 : Stern's diatomic series https://oeis.org/...
7 votes
2 answers
453 views

How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
1 vote
0 answers
58 views

Finite $p$-groups of co-class $3$, class at least $4$ and some controlled generator growth

I am trying to prove the following comment (Ref. https://link.springer.com/article/10.1007/s00605-016-0938-5 Page-684, Rmk3.2): Let $G$ be a finite $p$-group of co-class $3$, class $\geq 4$. Then $G$ ...
7 votes
1 answer
348 views

For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...
11 votes
1 answer
232 views

Does $\chi(1)^2=|G:Z(G)|$ for irreducible character of a finite group $G$ imply $G$ is solvable?

In "Character Theory of Finite Groups" I.M. Isaacs mention the following conjecture: It is only possible in a solvable group $G$ to have $\chi(1)^2=|G:Z(G)|$ with $\chi \in$ Irr$(G)$. Is this ...
4 votes
1 answer
358 views

Non-existence of projective covers

I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at: http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf In ...
13 votes
2 answers
2k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
8 votes
1 answer
203 views

Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?

Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-...
4 votes
1 answer
334 views

Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable?

I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book Greco, Silvio, ...
8 votes
2 answers
1k views

In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...
1 vote
0 answers
132 views

Terminology for representation all of whose isotypic pieces are nontrivial

Let $V$ be a finite-dimensional representation of a finite group $G$. Is there an adjective describing those $V$ for which every irreducible representation of $G$ is a direct summand of $V$?
10 votes
1 answer
533 views

Finite groups with automorphism mapping $a/b$ of the elements of $G$ to their own inverses? Case $a/b=3/4$?

I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question: name a group and an automorphism that takes $3/4$ of the elements of the group ...
7 votes
0 answers
241 views

Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$

$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...
27 votes
1 answer
2k views

Multiplying all the elements in a group

Let $G = \{ g_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is $$ G! = [ \prod_i g_{\sigma(i)...
6 votes
1 answer
214 views

Inner automorphisms of group algebras vs. inner automorphisms of the group

In a recent question on MSE I asked about conditions under which the canonical morphism $Out(G) \to Out(k[G])$ is injective. Is it true that this morphism is indeed injective if $G$ is finite and $...
1 vote
1 answer
261 views

Adding $n$-tuples over groups

Consider a finite abelian group $\mathcal{G}$. Let $S_0$ be a $n$-tuple of elements of $\mathcal{G}$, and let $S_i$ be the cyclically shifted version of $S_0$ by $i$ indices to the right. So for ...
5 votes
3 answers
207 views

Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$

Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index ...
-1 votes
1 answer
224 views

Normalizer of a Normalizer of a subgroup of a finite group with no elements of order $p^2$

Coming from a non-group theory background, I noticed that the finite groups I was dealing with seem to all have the following property. Let $G$ be a finite group, $H$ a subgroup. Then the normalizer $...
6 votes
1 answer
1k views

Concrete formula for Shapiro's Lemma

I wonder if there is a concrete formula to express the isomorphism in the well known Shapiro's Lemma that $H^i(G, \text{CoInd}_{H}^{G}(M)) \simeq H^i(H, M)$, where $H \subset G$ is a subgroup of $G$, $...
43 votes
3 answers
10k views

Feit-Thompson theorem: the Odd order paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...
2 votes
0 answers
200 views

Representation theory of 3-step nilpotent finite groups

I am interested in understanding the representation theory of certain finite nilpotent groups (over the complex numbers). The groups $G$ of interest have the following properties: 1) G is $3$-step ...
0 votes
1 answer
143 views

Probability distribution of random products of elements of a generating set of a finite non-abelian group

Let $G$ be a finite non-abelian group, and consider a choice of $N$ distinct elements $g_{0},g_{1},\ldots,g_{N-1}\in G$ that generate $G$. Now, let $t$ be an arbitrary positive integer, and let $d_{1},...
0 votes
1 answer
140 views

Lower bound for $[ H : H \cap xHx^{-1} ]$

Let $H$ be a subgroup of a finite group $G$, and let $N = N_G(H)$ be the normalizer of $H$ in $G$. For $x \in G$ is there a lower bound for $[ H : H \cap xHx^{-1} ]$?  If $x \in N$ this index is 1, ...
5 votes
1 answer
323 views

$C_4\times C_2 : C_2$: what does this mean?

I am reading this paper where the object $C_4\times C_2 : C_2$ is used as a group structure. I know that $C_n$ is a cyclic group but don't know what kind of operation between groups is identified by ...
4 votes
1 answer
185 views

Automorphism groups of a group and of its Fitting subgroup

This was a comment to the answer here . It is one of the series of questions about finite groups with automorphism groups of odd order and would reduce the question to nilpotent groups. Question. Is ...
2 votes
1 answer
971 views

Viewing a finite group as a group scheme

I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a ...
0 votes
1 answer
428 views

finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$. Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that $...
66 votes
1 answer
6k views

Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ...
4 votes
0 answers
119 views

Systems of imprimitivity for irreducible subgroup of GU(n,q)

My question is similar to this one but about finite field case. So, the set up is the following: Let $G$ be $GU_n(q)$ acting on unitary space $(V, {\bf f})$, where $V=\mathbb{F}_{q^2}^n$ and ${\bf f}...
5 votes
2 answers
215 views

Obtaining quiver and relations for finite p-groups

Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)? Since $KG$ is local, the quiver should ...
9 votes
0 answers
429 views

Which finite solvable groups have solvable automorphism groups?

Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable? The central case to deal with is that in which $G$ is a $p$-group of ...
62 votes
1 answer
4k views

Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$. ...
6 votes
0 answers
114 views

Schur indices for 2-groups

I am looking for any results on Schur indices over $\mathbb{Q}$ for 2-groups. By a theorem of Roquette (corollary 10.14 in Isaacs) these numbers are at most 2. I am interested in 2-groups for which ...
11 votes
1 answer
766 views

Automorphism groups of odd order

This is inspired by this question. Is there a description of finite groups without automorphisms of order $2$?
7 votes
1 answer
274 views

Groups and graphs

Let $A=(V, E)$ be a finite simple (no loops or multiple edges) graph. Let $G(A)$ be the following nilpotent group of class 2 and exponent $p$ (an odd prime). $G(A)$ as a set is $span(V)+span(E)$ ...

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