Questions about the branch of abstract algebra that deals with groups.

**2**

votes

**1**answer

49 views

### Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of ...

**22**

votes

**2**answers

436 views

### Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?

Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated ...

**8**

votes

**0**answers

71 views

### Are all sneaky groups products of Frobenius and 2-Frobenius groups?

I've been stuck thinking about this for a while.
Def. Let $G$ be a finite solvable group whose order is divisible by only three primes: $p,q,$ and $r$. Suppose that $G$ has cyclic subgroups of ...

**30**

votes

**0**answers

376 views

### Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : number of groups of order $n$
Does $N(n)=n$ hold for some $n>1$ ?
I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range ...

**7**

votes

**2**answers

459 views

### Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[[2, 0, 1, 7], [3, 4, 5, 6]]
[[4, 0, 6, 7], [2, 3, 1, 5]]
[[0, 7], [4, 6], ...

**3**

votes

**0**answers

92 views

### Wreath product of an abelian group with a nilpotent group

By work of Coulbois, the wreath product of two finitely generated free abelian group is $LERF$; i.e, every finitely generated group of this wreath product is closed in the profinite topology.
Is there ...

**5**

votes

**1**answer

122 views

### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...

**-6**

votes

**0**answers

78 views

### Show that $G$ a group? [on hold]

Suppose that $G$ is a semigroup , and for every $a$ in $G$ there is unique $a^*$ in $G$ that
$aa^*a=a$
Prove that $G$ is a group.

**4**

votes

**0**answers

109 views

+50

### Square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...

**14**

votes

**1**answer

314 views

### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...

**3**

votes

**2**answers

277 views

### Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated?
In view of Ian Agol's answer, I ...

**5**

votes

**1**answer

187 views

### number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere...
EDIT I am actually more interested in the number of conjugacy ...

**0**

votes

**0**answers

56 views

### Wild automorphisms of profinite groups

Is there a profinite group $G$, a continuous automorphism $\alpha$ of
$G$ and a topologically finitely generated closed subgroup $H \leq G$
such that $\alpha(H) \lneq H$ ?
Note that if an ...

**1**

vote

**0**answers

30 views

### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?

**3**

votes

**1**answer

170 views

### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by generators.
...

**7**

votes

**1**answer

143 views

### Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square.
To be exact, the group $G$ has a special element $1$, a unary ...

**1**

vote

**0**answers

69 views

### The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = ...

**1**

vote

**1**answer

115 views

### Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...

**4**

votes

**2**answers

121 views

### “Natural” action of a semidirect product of groups

Suppose we have a group $N$ acting on a set $X$, and a group $Q$ acting on $Y$. Moreover, we are given $\phi: Q \to \mathrm{Aut}(N)$.
My question is, is there a "natural" action on a set that can be ...

**5**

votes

**0**answers

146 views

### Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) ...

**1**

vote

**0**answers

89 views

### Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable.
Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...

**1**

vote

**0**answers

75 views

### Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated?
A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or ...

**5**

votes

**1**answer

129 views

### Upper bound on level of a congruence subgroup of the modular group

Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of ...

**9**

votes

**0**answers

159 views

### Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...

**5**

votes

**1**answer

208 views

### abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...

**11**

votes

**0**answers

183 views

### In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?

First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...

**2**

votes

**1**answer

105 views

### Is the Frattini subgroup of a free profinite group trivial?

Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial?
I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the ...

**18**

votes

**2**answers

970 views

### Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...

**1**

vote

**0**answers

94 views

### Centerless finite groups with exactly three prime divisors

Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups.
Is there any classification of such groups?
In what ...

**-8**

votes

**0**answers

50 views

### connectedness in Euclidean Space [closed]

f:[0,1] to R is a map defined by f(x)=sin(1/x),for x is not zero and
f(0)=0.X={(a,f(a))|a is in [0,1]}.prove that,X is a connected subset of
R^2 where the metric is usual euclidean metric on R^2.

**2**

votes

**0**answers

111 views

### Conjugate transpose and discreteness, for Kleinian groups

Let $G=\langle g_1,\dots g_n\rangle<\mathrm{PSL}_2(\mathbb{C})$ be discrete,
i.e. a finitely generated Kleinian group.
Let $H=\langle g^{\dagger}g\mid g\in G\rangle$,
where $g^{\dagger}$ is the ...

**3**

votes

**1**answer

123 views

### projective representation of supergroup

In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group ...

**7**

votes

**1**answer

193 views

### Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...

**3**

votes

**0**answers

114 views

### Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:
(1) On the probability of satisfying a ...

**34**

votes

**1**answer

3k views

### Infinitely many solutions of a diophantine equation

If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...

**3**

votes

**1**answer

157 views

### An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group.
(i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$).
Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ?
Here, $\beta_{2}^1(G)$ stands ...

**6**

votes

**1**answer

262 views

### When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...

**1**

vote

**2**answers

139 views

### Abelian group of finite rank

Let given torsion free abelian group $A$ of finite rank. Let for prime number $p$, given that $\cap_i p^iA =\{0\}$. Is it true that for any $p$- torsion abelian group $B$, $\text{Hom}_{\mathbb{Z}}(A, ...

**-6**

votes

**1**answer

286 views

### Quintic Equation [closed]

Can we solve the following polynomial quintic equation by radicals
x^5 + x^4 = 1
I found one real root which is algebraic solution (no approximation method ...

**10**

votes

**2**answers

350 views

### minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...

**0**

votes

**0**answers

164 views

### Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...

**11**

votes

**5**answers

2k views

### Finite groups with elements of the same order

Given a finite group $G$, let $\{(1,1),(m_1,n_1),\ldots,(m_r,n_r)\}$ be the list of pairs $(m,n)$ in which $m$ is the order of some element, and $n$ is the number of elements with this order. The ...

**27**

votes

**2**answers

3k views

### No simple groups of order 720?

I think most students who first learn about (finite) groups, eventually learn about the possibility of classifying certain finite groups, and even showing certain finite groups of a given order can't ...

**5**

votes

**1**answer

171 views

### Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...

**3**

votes

**2**answers

139 views

### Automorphism group of a free product

Suppose that $G$ and $H$ are groups (not isomorphic) and $G\ast H$ the free product. Let $Aut(G)$, $Aut(H)$ be the automorphism groups of $G$ and $H$. What is $Aut(G\ast H)$ ?

**17**

votes

**1**answer

507 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...

**12**

votes

**1**answer

306 views

### Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite

I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their ...

**4**

votes

**1**answer

175 views

### when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...

**2**

votes

**0**answers

100 views

### Is a finitely generated residually free group “almost LERF”?

Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
...

**8**

votes

**2**answers

316 views

### Characterization of finite groups using sum of the orders of their elements

Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$.
Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...