Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
370 questions
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Minimal irrep of $\mathrm{PSL}(2,p) $
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ p $ be a prime for which $ \PSL(2,p) $ is simple (so $ p \neq 2,3 $).
Is the ...
- 4,081
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1
answer
357
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Alternating subgroups of $\mathrm{SU}_n $
$\DeclareMathOperator\PSU{PSU}$Let $ \PSU_n $ be the projective unitary group. Let $ A_m $ be the alternating group on $ m $ letters.
$ A_5 $ is a maximal closed subgroup of $ PSU_2 \cong SO_3(\mathbb{...
- 4,081
11
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1
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739
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Three involutions on the set of 6-box Young diagrams
The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...
- 22.3k
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4
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2k
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Iterated semi-direct products
Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where $...
- 1,661
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4
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Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
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2
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932
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A group action of the Heisenberg group with special symmetries
Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
11
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2
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Aut(G) = $C_3$, G = ?
Is there a group G such that Aut(G) = $C_3$? What if we replace 3 with a prime number p?
- 423
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When is the semidirect product of an elementary abelian group and a cyclic group generated by two elements?
I am trying to characterize when a semi-direct product of the form $(Z/pZ)^n \rtimes (Z/qZ)$ is isomorphic to a group generated by two elements. Here $p$ and $q$ are distinct odd primes.
I would be ...
- 2,720
10
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3
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Low-dimensional irreducible 2-modular representations of the symmetric group
I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
- 1,298
10
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1
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Generalizing the Notion of Nilpotent/Abelian/Cyclic Numbers
Let us call a number $n\in\mathbb{N}$ nilpotent if
$$n=p_1^{e_1}\cdots p_m^{e_m}$$
with $p_i^k\not\equiv 1\mod p_j$ for $i,j\in\{1,\ldots,m\}$ and $1\leqslant k\leqslant e_i$.
A cute theorem says the ...
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5
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Reference requested: Random walk on groups
I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...
- 101
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2
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547
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Arbitrarily large finite irreducible matrix groups in odd dimension?
I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an ...
- 13.6k
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Permutation groups with diameter $O(n \log n)$
I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture:
Suppose that
1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
- 7,545
10
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1
answer
672
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Homomorphic images of a Cartesian product of finite groups
What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...
- 2,804
10
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2
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Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4
I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
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0
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445
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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 ...
- 44.4k
9
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0
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165
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When is the rank 2 free metabelian group of exponent $n$ center free?
Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
- 7,527
9
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1
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292
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Nonabelian finite groups with "locally commuting" presentation
Let $G = \left\langle S | R \right\rangle$ be a finitely presented group where S is a set of generators and R is a set of relations. We say that the presentation is "locally commuting" if whenever two ...
- 141
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Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
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Equivariant cohomology of finite group actions and invariant cohomology classes
Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an ...
- 445
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What classes of groups can arise as "symmetry groups of terms"?
Let $\mathfrak{A}$ be an algebra (in the sense of universal algebra). To each term $t(x_1,...,x_n)$ in the language of $\mathfrak{A}$ in which each variable actually appears we can assign a group $G_\...
- 21.2k
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2
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2k
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Which finite groups are generated by n involutions?
One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...
- 1,861
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1
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482
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The degree of a constant polynomial on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
- 41.8k
9
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0
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297
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An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
9
votes
2
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525
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Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...
9
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2
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843
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Formula for the Frobenius-Schur indicator of a finite group?
Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.
Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^*$, then $V$ admits a ...
- 2,821
8
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2
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617
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sum-sets in a finite field
Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.
Question. Is it true there is always a $\pi\in\...
- 43.2k
8
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1
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495
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What is the Schur multiplier of the affine linear group AGL(n,q)?
What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation ...
- 533
8
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2
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586
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How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?
Let $G \leq {\rm S}_n$ be a finite permutation group, and let
$S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The growth ...
- 19.6k
8
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2
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799
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characters on a finite group with `extremal' behaviour
The following question is a bit technical, and I haven't got to grips with it enough to be able to present it as a well-focused question. However, my hope is that the collective group-theoretic ...
- 25.8k
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4
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Bounds on number of conjugacy classes in terms of number of elements of a group ?
What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...
- 375
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1
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Finite groups containing no subgroups of a given order or index
The classical Lagrange's Theorem says that the order of any subgroup of a finite group divides the order of the group. For abelian groups this theorem can be completed by the following simple fact: ...
- 41.8k
8
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1
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256
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Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation
In this post, irrep and dim mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) ...
8
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1
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522
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Trivial group cohomology induces trivial cohomology of subgroups
From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
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2
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600
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How many minimum generating sets are there in a finite group?
Let $G$ be a finite group of order $n$.
A generating set in $G$ is said to be minimum if it has minimal size.
Is there a known lower bound on number of minimum generating sets in a group of order $...
user108347
8
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2
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576
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Two statistics on the permutation group
Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets
$$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\...
- 43.2k
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0
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666
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Approximating Lie groups by finite groups
How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$...
- 8,086
8
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1
answer
562
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The parity of the full automorphism group order of finite non-abelian groups of prime exponent
Is there a finite non-abelian group $G$ of prime exponent such that the full automorphism group of $G$ is of odd order?
- 3,738
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304
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A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
8
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4
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658
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Normal Covering of a Finite Group
Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
7
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3
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650
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Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
- 5,230
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1
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Are the distributive permutation groups linearly primitive?
An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no non-...
7
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2
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780
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Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
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1
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356
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Number of conjugacy classes of pairs of commuting elements II
This post follows up on a discussion initiated in Number of conjugacy classes of pairs of commuting elements.
Consider a finite group $G$ and let $r_G$ represent the number of conjugacy classes of ...
7
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2
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422
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Automorphism group of a special commuting graph
Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets ...
- 81
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Irreducible representations and Jaccard Kernel for Groups?
Since each group $G$ can be considered as a subgroup of the symmetric group, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
$$X(\pi) := \{ (i,\pi(i)) |...
user6671
7
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1
answer
568
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recognition of symmetric groups in GAP
In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes ...
- 16.9k
7
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4
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768
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How many finite simple groups of order $p+1$?
I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number.
But they don't seem to fall into any classification - have these all been determined? Is the number of them even ...
- 1,180
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2
answers
764
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Proofs of a character identity?
Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...
- 5,349
7
votes
2
answers
456
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Subgroup property stronger than being characteristic
In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...
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