# Questions tagged [finite-groups]

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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### Sum of all elements of a $p$-section of a group

Let $G$ be a finite group. A $p$-section of $G$ is a subset of the form $S(x) = \{g \in G \mid g_p \text{ is conjugate to } x\}$, where $x$ is a $p$-element of $G$, with $p$ a prime, and $g_p$ denotes ...
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### Question about a possible application of Burnside's basis theorem

Let $G=\langle a_1,\dots , a_d\rangle$ be a $d$-generator $p$-group of order $p^n$ (i.e. $d$ is the minimal number of generators). Further let $N$ be a characteristic and elementary abelian subgroup ...
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### Does an irr. rep. of finite $G$ have a basis of the form $\{hv : h \in H\}$ for $H$ subgroup of $G$?

Let $G$ be a finite group and $V$ be a (complex vector space) representation of $G$. Consider the following three facts: Whenever a $V$ is an irreducible representation of $G$, the dimension of $V$ ...
51 views

### How to check whether a finite $p$-group is regular in GAP?

I am trying to check whether a given $p$-group is a regular $p$-group in GAP. I am trying to use the command 'IsRegularPGroup(G)' for it. However I am getting 'Error, Variable: 'IsRegularPGroup' must ...
• 354
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### Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
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### Brauer Tables for the Monster Group

I am doing research on modular character theory and wanted to study the Brauer Table for some of the larger simple groups, however I have been struggling to find the Brauer Tables for the monster ...
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### Criterion for cyclic groups in terms of its number of subgroups

Let $G$ be a finite group of order $n$. Suppose that for every other group $H$ of order $n$, the number of subgroups of $H$ is at least as big as the number of subgroups of $G$. Does it imply that $G$ ...
203 views

### A group with exactly half of the elements in one conjugacy class

Suppose $G$ is a finite group and there exists a conjugacy class $S$ in $G$ contains exactly $|G|/2$ many elements. What can we say about $G$? For example if $G$ is a Dihedral group $D_{n}$ with $n$ ...
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### Derived series of a square-free order group stabilizes

I have been studying from these notes on groups, rings and fields by Lenstra and I find myself struggling with problem 1.20 which states the following Let $G$ be a finite group of squarefree order. ...
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### Possible indices of finite index subgroups of $SL_2(\mathbb{Z})$

Here are 3 basic observations regarding $SL_2(\mathbb{Z})$: The abelianization of $SL_2(\mathbb{Z})$ is isomorphic to $\mathbb{Z}/12\mathbb{Z}$ and so $SL_2(\mathbb{Z})$ has a finite index subgroup ...
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### Let $G$ be the special linear group $SL_2(3)$ [closed]

Let $G$ be the special linear group $SL_2(3)$, i.e., the set of all $2 \times 2$ matrices with coefficients from the field $\mathbb{Z}_3$ and determinant equal to 1, with the operation of ordinary ...
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### Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants [closed]

Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants (line with equation $y = x$), and let $b$ be the reflection of the same plane over the bisector of the even ...
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### Basis set of a group ring [closed]

Let $(G,\cdot)$ be a finite abelian group and $\mathbb{Q}G=\{\sum {r_g}g:r_g\in \mathbb{Q}, g\in G\}$ be a group ring of $G$ over the rational numbers $\mathbb{Q}$. I would like to know how a basis ...
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### Group of order $n$ is a subgroup of $S_{n-1}$

Let $G$ be a group of order $n$, where $n>1$ is not a prime power. Show that $G$ is a subgroup of $S_{n-1}$. I believe I will need to use some form of Cayley's Theorem and/or Cauchy's Theorem, but ...
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### Working with character tables

I am currently a bit stumped by an old exam question, which gives a character table and wants me to deduce properties of the group: What is the order of $g$? Show that $g \notin C_G(G)$ Show there ...
1 vote
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### Normalizer of a Young subgroup in symmetric group

In $S_n$, a Young subgroup $S_{r_1}^{m_1}\times S_{r_2}^{m_2}\times ... S_{r_k}^{m_k}$ where $m_1\times r_1+...+m_k\times r_k=n$ has normalizer $N=S_{r_1}wr S_{m_1} \times ...\times S_{r_k}wr S_{m_k}$....
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### How to prove that all elements inside a cycle of a cyclic group are different from each other [closed]

Let $G$ be a finite cyclic group $(G, \circ)$ and $a \in G$: $$\langle a \rangle = \{a^z : z \in \mathbb{Z}\},$$ and $\operatorname{ord}(a) = \min\{a^n : n \in \mathbb{N}_+\}$, i.e., the smallest ...
1 vote
54 views

### IsGroupOfAutomorphisms functionality

I'm looking for two things related to the GAP function IsGroupOfAutomorphisms: whether it does what I think it does (based on the brief GAP manual entry), and if so, how it works. The GAP manual ...
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### meaning of $IBr(X | Q)$

In the paper 1, there is a notation used without specifying the meaning. It is $IBr(X | Q)$ in Definition $4.1$. What it means? Irreducible Brauer characters of the group X from a block with defect ...
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### Schur’s lemma over $\mathbb{F}_p$

I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement. Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
364 views

### No simple group of order 756 : Burnside's proof

I'm interested in a proof of the non-simplicity of groups of order 756. W.R. Scott, Group Theory, p. 392, exerc. 13.4.9, gives it as an easy exercise, but depending on rather advanced results. I have ...
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### Invariants of $2$-torsion group under involution

Let $G=\{1,\tau\}$ be the group with two elements and let $A$ be a free abelian group of finite rank on which $G$ acts (via group homomorphisms). Let $B$ be a $2$-torsion group, also with an action ...
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### Which transitive $G$-sets appear when repeatedly inducing and restricting $G/H$, where $H\subseteq G$ is an inclusion of finite groups?
Let $G$ be a finite group and $H$ be a subgroup. Then $G/H$ is a transitive $G$-set. We can define a sequence of $G$-sets as follows: $$X_0=G/H$$ $$X_{n+1}=Ind_H^GRes^G_HX_n$$ Like every finite $G$-...