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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Irreducibility of a certain polynomial associated to an irreducible representation of a finite group

Let $k$ be an agebraically closed field of characteristic 0. Let $G$ be a finite group of order $n$. A representation of $G$ is a homomorphism $\psi: G \rightarrow GL(V)$ where $GL(V)$ is the general ...
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69 views

Maximal abelian subgroup of general linear groups

Thanks for any help or comments. Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$. By maximal abelian I mean if $A$ is ...
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76 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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40 views

Can $G$ be a union of some of$H$ copies?

Let $G$ be a finite group and $H$ be a proper subgroup. Then we know that $$G\neq \bigcup_{g\in G} H^g$$ I wonder whether this is possible $$G= \bigcup_{\sigma \in Aut(G)} \sigma (H)$$ If there ...
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104 views

$\gcd(|G|, |\text{Aut}(G)|)=1$ means G is abelian?

Prove the following assuming that $G$ is finite group with $\gcd(|G|, |\text{Aut}(G)|)=1$. a) G is abelian (done). b) Every Sylow subgroup of $G$ is cyclic of prime order. Since G is ...
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64 views

Group of order 112

Let $G$ be a finite group of order $2^4\times 7$ and Sylow $7$-subgroup of $G$ is not normal. Prove that Sylow $2$-subgroup of $G$ is abelian. I am very grateful for any help in this problem.
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75 views

a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
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Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, ...
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126 views

Galois group of $x^8-2$ and intermediate fields

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to ...
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52 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
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Permutation group of a set

If we let $G$ be a finite permutation group of a finite set $X$ and assume that $G$ has exactly 2 orbits of the same cardinality, how can we show that there is some permutation in G that has no cycles ...
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45 views

On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{q})$

This question has already been asked here. However, the answer there isn't particularly helpful. Consider $G = GL_{n}(\mathbb{F}_{q})$ where $q = p^{r}$ where $p$ is prime. Show that the upper ...
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About motivation of Lang's Proof $S_n$ is not solvable for $n\geq 5$.

In Lang, he proves that $S_n$ is not solvable if $n\geq 5$ by using following observation. If $N\unlhd H\leq G$, H contains every 3-cycle, and if $H/N$ is abelian, then H contains every 3-cycle. Where ...
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Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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47 views

Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
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108 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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How can we find element orders of a finite group?

Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please ...
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Simple question on Sylow subgroups.

Given a finite group $G$, $|G|=p^am$, $P,Q\in\operatorname{Syl}_p(G)$, then we know that $Q=P^x$ for some $x\in G$. Thus every element of $Q$ is the image of an element of $P$ under the isomorphism ...
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51 views

Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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Constructing groups from actions

As an example to set the scene, suppose I have a cube and I let its symmetry group act on its faces. Every face admits four transformations which leave the cube invariant and which maps the chosen ...
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Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq ...
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63 views

What is the difference between operator groups and group actions?

I have always thought that operator groups and group actions are two names for the same thing. Now I have noticed that they have different codomains. Actually I am still confused, why are they ...
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126 views

Central extensions of elementary abelian p-groups

Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is : How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n ...
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108 views

On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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Groups of order 8

Ok so I am looking at a proof to all the groups of order 8, I've attached an image which basically is the start of the proof. In particular the part highlighted with yellow is causing me problem, I ...
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Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
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230 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
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An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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why does table look like this? - multiplicative table for Group

I have a Group $\mathbb{Z_2}[x]/q$ where $q(x)=x^3+x+1$ In my textbook, there a multiplication table: but I dont understand how the last $x+1$ comes. it should be $(x^2+x+1)*(x^2+x+1) \quad mod ...
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Subgroups of Cyclic $p$-Groups

Theorem Let $G$ be a cyclic group of order $n$, generated by $x$. Then every subgroup of $G$ is cyclic. The proof is as follows: let $H$ be a proper non-trivial subgroup. Then there exists a ...
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Partition of Symmetric Group

For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...
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63 views

On the Example of J. Alperin

In the paper "Large Abelian Subgroups of $p$-Groups" by J. Alperin, the author constructs an example of a $p$-group of order $p^{3n+2}$ ($p>2$) in which any abelian subgroup has order at most ...
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Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
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110 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
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163 views

Is there a simple way to show that wreath product is associative?

Is there a simple way to show that wreath product is associative? If your proof is short, please write it explicitly. Thank you.
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87 views

Equality in commutator subgroup

I will start by apologizing if this questions seems twisted. I am reading the paper Cohomology theory of groups with a single defining relation (Lyndon, $1950$) and the question comes from page $659$ ...
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Subgroups of $G^n$

Let $G$ be a non-Abelian group and let $n$ be a (large) positive integer (eventually going to infinity). For simplicity, let's take $G=\mathbb{D}_6$ (Dihedral group of order $6$). Is there a way to ...
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Commutators in finite groups

I get stuck with this question : Let $g$ be an element in a finite group $G$, and let $k$ be an integer coprime to $|g|$. Prove that $g$ is a commutator in $G$ if and only if $g^k$ is a commutator. ...
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174 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
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Conjugacy classes in $SL(n,q)$

If $q$ is a prime power, then the conjugacy classes in $GL(n,q)$ are determined completely by minimal polynomials. Now in the subgroup $SL(n,q)$, these conjugacy classes can split, in the sense they ...
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267 views

On Sylow subgroup of simple group PSL(2,p)

Let $G$ be finite group such that $Z(G)=1$ and the number of Sylow $p$-subgroups of $G$ equal to the number of Sylow $p-$subgroups of $PSL(2,r)$ for every prime $p$ ( $r$ is prime and $r^2$ not ...
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169 views

About a Sylow subgroup of a product

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of G such that $P=(P\cap H)(P\cap K)$. I looked at the proof here, but I ...
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A subclass of metacyclic groups

A group $G$ is metacyclic if it contains a cyclic normal subgroup $N$, such that $G/N$ is also cyclic. $N$ is called the kernel, and the order of $G/N$ is the index of the metacyclic group. There is ...
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A class of finite groups

Let $G$ be a finite group having the property that for any prime $p$ dividing $|G|$, it has a subgroup H with $[G:H]=p$. What can be said about these groups? I believe I can prove that they must be ...