Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
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128 views

References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
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183 views

Values attained by $|G/Z(G)|$?

So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I ...
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302 views

Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ...
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182 views

How confident can we be about the validity of the classification of finite simple groups?

The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed. How certain can we be that The proof of ...
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293 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
9
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79 views

if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
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127 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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166 views

A question about sylow subgroups and $C_G(x)$

Let $G=PQ$ where $P$ and $Q$ are $p$ and $q$-sylow subgroups of $G$ respectivly. In addition suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$ where $Z(G)$ is the ...
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119 views

Central Quotients of Finite Groups

There are more than 50 groups of order 48, and among them 16 groups have center of order 2, let $G$ be among such groups. Then $G/Z(G)$ is a group of order 24. What is this group of order 24? There ...
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117 views

Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
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142 views

Official name(s) for a certain type of p-group

I'm implementing a class of groups into Sage (sagemath.org), a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called ...
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124 views

How useful are geometric aspects when studying finite groups?

My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
8
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193 views

Generating the partners in a multi-dimensional irreducible representation.

I am trying to block diagonalize a Hermitian matrix using the irreducible representations of its symmetry group. Using the group's character table, it is straightforward to generate a set of ...
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88 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
7
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159 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...
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54 views

Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?

If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ ...
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92 views

Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
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156 views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
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71 views

Fixed points of coset operation

Let $G$ be a finite group which operates on two finite sets $E_1$ and $E_2$. Say that $E_1$ and $E_2$ are weakly $G$-isomorphic if for every $g \in G$, $\mathrm{Card}(E_1^g)=\mathrm{Card}(E_2^g)$, ...
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257 views

What groups are semidirect products of simple groups?

The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under ...
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118 views

character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here: Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in \operatorname{Irr}{(G)}$? Can ...
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93 views

Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with 2 generators. More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of ...
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146 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
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228 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
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287 views

Resource: Group Theory

There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory. I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
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181 views

A problem in transfer theory

This is about problem 5B.1 page 157 in Isaacs' "Finite Group Theory" book. This chapter is definitely giving me trouble. The problem reads: Let $G$ be a finite group, $P \in \operatorname{Syl}_p(G)$ ...
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125 views

The order of $H$ is relatively prime to its index $[G:H]$

Suppose that a subgroup $H$ of a finite group $G$ satisfies one of the following two conditions: (i) For any nonidentity element $x$ of $H$ we have $C_{G}(x) \subset H$ (ii) If $K$ is a subgroup of ...
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185 views

Show any group of order $275$ has an element of order $5$.

This is what I have. Note: I'm not allowed Cauchy's theorem or Sylow theorems. Let $|G| = 275$. So I know $275 = 5\times5\times11$. If I assume that $G$ is cyclic then there exists $x\in G$ such that ...
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301 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
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105 views

Is there a classification of finite abelian group schemes?

There is a well-known classification of finite abelian groups into products of cyclic groups. What about finite abelian group schemes, where we may put in the qualifiers "affine", "etale", or ...
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89 views

$H=\langle a,b| a=bab, b=aba\rangle $ and $\frac{H}{A}\cong\ Q_8$

Here is my problem: Let $$H=\langle a,b| a=bab, b=aba\rangle $$ and $\frac{H}{A}\cong\ Q_8$ wherein $A\leq Z(H)\cap H'$. Show that $H\cong Q_8$. Working on the elements, I could see that ...
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100 views

Possible subgroups of $\mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$

$G \cong \mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$ $H\leq G$ so that $G/H \cong \mathbb{Z}/3^2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z} $ Find all possible ...
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198 views

Icosahedral symmetry as permutation group

Hopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the ...
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248 views

Classifications of finite nilpotent groups

I would like to understand the concept of classification in the context finite groups. For finite abelian groups and finite simple groups, it's clear to me what is meant by classification. However, ...
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67 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
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66 views

Question about inverse galois problem

I had a question… if for every finite simple group, we can construct a galois extension over Q with that galois group…. does it follow that we can construct galois extensions over Q with any finite ...
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33 views

Visual Solution - Find All (Cyclic) Subgroups of $D_4$ generated by 1, 2, … elements - Fraleigh p. 84 8.19

Verify that the subgroup diagram for $D_4$ shown in Fig. 8.13 is correct by finding all (cyclic) subgroups generated by one element, then all subgroups generated by two elements, etc. Here, $p_i$ mean ...
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102 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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78 views

on the simple group $M_{11}$

As we know, the simple group $M_{11}$ is a important group,it has order $7920$, how can we prove the simple group of order $7920$ is isomorphic to $M_{11}$ ?
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What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
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142 views

Examples of finite groups that are not a semidirect product

I'm looking for examples of (families of) finite groups that are not semidirect products. When first learning group theory, the first such group that one encounters is $Q_8$. In my search for other ...
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112 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
4
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92 views

Prime divisor in the Automorphism group

Suppose that $G$ is a non-abelian simple group and $r$ is a prime number such that $r$ not divide $|G|$. It is a question for me that whether $r$ can divide |Aut($G$)|?
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158 views

Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
4
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0answers
147 views

Easy proof of trivial fusion implies normal p-complement

Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup If A and ...
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146 views

A problem of J.D. Dixon

Referring to his paper from 2004, I was wondering if anyone is aware of any relevant work done on the following problem: Of course, the case $w(X_1,X_2)=X_1X_2X_1^{-1}X_2^{-1}$ admits the answer ...
4
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129 views

Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$. Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a ...
4
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0answers
83 views

Primitive permutation group with subdegree 4

What are the possible sizes of a point stabilizer in a primitive permutation group, where the point stabilizer has an orbit of size 4? In the tradition of subdegree 3 and subdegree 2, I wonder ...
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159 views

Is $H^2$ weakly closed in $H$? Isaacs Finite Group Theory, exercise 8B.6

I'm trying to solve exercise 8B.6 on page 249 of Isaacs's Finite Group Theory textbook (the second question in a series; this is the third as question here). I have an idea, but it doesn't quite ...