Questions on group theory which concern finite groups.

**4**

votes

**0**answers

57 views

### Examples for Bogomolov multiplier of finite group

Take $G$ to be a finite group. The projective representations of $G$ are classified by the group cohomology $H^2(G,U(1))$. We focus on the restriction map
$H^2(G,U(1)) \xrightarrow[]{\rm restriction} ...

**1**

vote

**0**answers

70 views

### What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is
$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$
$G$ transitive permutation group?
And what are the ...

**17**

votes

**0**answers

304 views

### Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?

Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements
and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$.
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...

**3**

votes

**1**answer

102 views

### On decomposition of finite Abelian groups

It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $...

**6**

votes

**1**answer

186 views

### Reference request: an elementary result on characters of finite abelian groups

The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $...

**4**

votes

**1**answer

173 views

### Determining the conjugacy classes of a wreath product $G \wr S_n$

If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S_n \cong G^n \rtimes S_n$ be determined?

**5**

votes

**1**answer

112 views

### Is there a subgroup of dual depth 3?

This post is motivated by an exchange with Zhengwei Liu. It is more than the dual version of this post, because we consider any subgroup (instead of just maximal), and even more at the end...
Let's ...

**9**

votes

**2**answers

420 views

### About normal minimal subgroups not in the Frattini

In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact.
$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among ...

**6**

votes

**3**answers

287 views

### Is there a maximal subgroup of depth 3?

Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...

**3**

votes

**0**answers

83 views

### Normalizer of a split torus

Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...

**7**

votes

**0**answers

164 views

### Number of elementary abelian subgroups of extraspecial $2$-groups

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following ...

**5**

votes

**2**answers

190 views

### Is there a size 2 generating set of the signed symmetric group $B_n$?

The signed symmetric group $B_n$ is a permutation group where the underlying set is $B_n=\{\sigma \in S_{A_n}| \forall x \in A_n, \sigma(-x)= -\sigma(x)\}$ with $A_n=\{-n,-(n-1),-(n-2),\cdots,-1,1,\...

**17**

votes

**2**answers

419 views

### What's the maximum probability of associativity for triples in a nonassociative loop?

In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...

**4**

votes

**1**answer

234 views

### A name for a group with finite abelization?

Let us recall that a group $G$ is called perfect if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial.
Question. Is there any name for a group ...

**7**

votes

**1**answer

291 views

### The probability that two elements of a finite nonabelian simple group commute

It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...

**5**

votes

**0**answers

108 views

### lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...

**6**

votes

**3**answers

141 views

### Groups whose poset of direct factors are lattices

Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...

**2**

votes

**1**answer

132 views

### Extensions of lattices

Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...

**1**

vote

**1**answer

183 views

### A permutation with reflection property

Consider permutations $\pi$ of the set $\{1,\dots,n\}$ having the symmetry property $\pi \pi^* \pi = \pi^*$, where $\pi^*$ is the "reflection" $k \mapsto n+1-k$. Are there references or other ...

**11**

votes

**1**answer

310 views

### How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2
Can this sentiment be made precise
in the sense of the
Turán and Erdős definition of "the probability that two elements of ...

**3**

votes

**1**answer

86 views

### Operation of a p'-group on a set of p-power order and fix points

The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....

**4**

votes

**1**answer

162 views

### a question on Deligne-Lusztig characters

Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be ...

**7**

votes

**1**answer

254 views

### Conjectured combinatorial non-equality

Let $n,k,\ell$ be integers for which $0\leq k<\ell \leq n-6$. For a fixed $n$, think of $k,\ell$ as being allowed to vary. I believe the values
$$(n-k-5)(k+1)(k+2)\binom n{k+3}~~~\text{and}~~~(n-...

**7**

votes

**2**answers

246 views

### What are the automorphism groups of direct products of dihedral group D4

What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?
For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(...

**2**

votes

**2**answers

206 views

### Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups

I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible ...

**3**

votes

**0**answers

158 views

### Given any group G, how can we construct another group H such that G is isomorphic to the commutator subgroup H' of H?

First of all, it's not true that any group can be realized as the commutator subgroup of some group. So, if we assume that there is atleast one group H with H' isomorphic to G, how to construct all ...

**0**

votes

**1**answer

135 views

### Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]

Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...

**3**

votes

**1**answer

143 views

### When is the semidirect product of $(Z/pZ)^n$ and $(Z/qZ)^2$ generated by two elements?

I asked a very similar question here and got a wonderful answer. But now I need to change the question slightly (this is the last question like this, I promise).
I would like to characterize when $(\...

**4**

votes

**0**answers

128 views

### Normalizers of abelian Sylows in simple groups

Suppose $G$ is a (nonabelian) finite simple group and $p$ is a prime such that the $p$-Sylow in $G$ is abelian. What can be said about its normalizer? I'm particularly interested in lower bounds on ...

**2**

votes

**1**answer

234 views

### Schreier conjecture — without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...

**0**

votes

**0**answers

54 views

### Solvability of finite group from indices of commutator and abelian normal subgroup

Suppose finite directly indecomposable group $G$ has $\frac {|[G, G]|}{|G|} < \alpha$ and $\frac {|A|} {|G|} > \beta$, where $A \lhd G$ abelian. Are there some nontrivial bounds on $\alpha, \...

**9**

votes

**1**answer

308 views

### 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 ...

**3**

votes

**1**answer

83 views

### Lower frattini subgroup

Let $G$ be a finite group. Define $\Phi_{-}(G)$ as the subgroup of $G$ generated by all the minimal subgroups of $G$ (a minimal subgroup of $G$ is a subgroup of $G$ of prime order).
It is easy to ...

**-1**

votes

**1**answer

86 views

### Is the numerable product of finite abelian groups a cantor set? [closed]

Today I have heard that statement but I can't find the reference.
Can somebody know a reference and/or a proof?

**7**

votes

**1**answer

200 views

### Finite group with a character having one nonzero absolute value

Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$.
...

**5**

votes

**1**answer

417 views

### Question on Hall's theorem

Theorem 9.3.1 in Hall's group
theory says: Let $G$ be a solvable group and $|G|=m\cdot n$, where $%
m=p_{1}^{\alpha _{1}}\cdot \cdot \cdot p_{r}^{\alpha _{r}}$, $(m,n)=1$. Let $%
\pi =\{p_{1},...,p_{r}...

**2**

votes

**0**answers

71 views

### Width of symmetric groups

MSE crosspost
For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...

**1**

vote

**0**answers

136 views

### about a strange property of p-groups of maximal class

I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property :
If s is an element in $G-G_1$ ($G_1$ is ...

**1**

vote

**0**answers

42 views

### Weighted cancellation norm of a word computation

A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$.
We say that two disjoint pairs $\{...

**1**

vote

**0**answers

84 views

### Certain $p$-group with cyclic center

Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group.
(i.e., possesses at least one non-normal subgroup).
Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order $...

**5**

votes

**2**answers

308 views

### $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?

Let $G$ be a finite group of order $240$.
If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are
$
[1,1,1,1,~3,3,3,3,3,3,3,3, ~...

**7**

votes

**0**answers

207 views

### Integral representations of finite groups and lattice point geometry

This contains both a reference request, and a specific problem.
Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider ...

**7**

votes

**0**answers

106 views

### Small modules over finite group with large cohomology

Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here.
Using Lyndon resolvent, ...

**4**

votes

**0**answers

176 views

### Conjugacy class representatives for the automorphism group of a finite abelian group

Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...

**3**

votes

**2**answers

354 views

### Finite groups with small God's numbers

Let $G$ be a finite group and $S$ be generating set it. Now given all words with alphabet $S$, then there exists a minimum word length $N(S,G)$ such that all group elements are represented by a word ...

**0**

votes

**1**answer

160 views

### Is $G$ non-solvable?

Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that
(1) there is a character $\chi\in\mathrm{...

**5**

votes

**0**answers

65 views

### A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...

**1**

vote

**0**answers

86 views

### Character degrees of a finite group?

Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...

**11**

votes

**1**answer

247 views

### Genus of Cayley graph of $A_5$ with two generators of order 5

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. I am wondering what is its graph genus (orientable or non-orientable). The best I could get by trial and error is an ...

**4**

votes

**0**answers

106 views

### Does $G$ have a normal abelian Sylow $2$-subgroup?

Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...