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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Finite matrix groups as subgroups of $S_n$.

I have heard that all finite subgroups are isomorphic to a subgroup of $S_n$. I was thinking about examples of this. In particular I would like to know how this works for certain matrix groups. The ...
2
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1answer
40 views

Existence of a unique subfield of degree dividing the degree of a given irreducible polynomial

Let $n$ be a positive integer and $d$ a positive integer that divides $n$. Let $\alpha \in \mathbb{R}$ be the root of the polynomial $x^{n} - 2 \in \mathbb{Q}[x]$. Prove that there is precisely one ...
2
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2answers
47 views

What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
3
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2answers
97 views

How did we classify the finite simple groups when we haven't classified the primes?

Why was the classification of the finite non-abelian simple groups "easier" (!!) than the classification of the finite abelian simple groups [the prime numbers], which still doesn't exist? (Does it? ...
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1answer
45 views

Prove that any group is a disjoint union of conjugacy classes.

How do I prove that any group is a disjoint union of conjugacy classes? Any reading reference would also be helpful
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1answer
82 views

How to read GAP's output on “IrreducibleRepresentations”?

For example for the group $SL_2(\mathbb{F}_3)$ I get the following, ...
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1answer
20 views

Finitely presentated subgroups of a group are normal?

If a group is finitely generated, then it is a quotient of the free group on the set of generators. Further if a subgroup of a group has some finite presentation, does it mean that it is normal, ...
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0answers
113 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
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1answer
39 views

Are these groups of order $81$ isomorphic?

The classification of groups of order $p^4$ is well known. However, different sources sometimes classify in different way. I am trying to compare, in the case $p=3$ between Burnside list (p. 100-102) ...
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1answer
48 views

Lifting representations, kernels and invariant subspaces

Let $G$ be a group, $N \triangleleft G$, $G/N$ the corresponding quotient group. Suppose $\rho : G/N \longrightarrow GL(\mathbb{C})$ is a representation of $G/N$. Then the composition ...
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2answers
50 views

Action of $G$ on the left cosets of $H$ giving a non-trivial homomorphism

If $H < A_5$ is a subgroup of index $3$, the action of $G$ on the left cosets of $H$ gives a non-trivial homomorphism $$\underset{order \ 60}G \rightarrow \underset{order \ 6}{S_3}$$ which ...
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1answer
52 views

Considering transitive $G$-set

Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ ...
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2answers
67 views

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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0answers
24 views

Inducing $A_4$ from $\langle (123) \rangle$

Let $G=A_4$ and $H=\langle (123) \rangle < G$. Compute $Ind_{H}^G \chi$ for every irreducible $\chi$ of $H$. Choose the right transversal of $H$ in $G$ as $V_4=\{ 1, (12)(34),(13)(24),(14)(23) ...
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3answers
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Proving that $D_{12}\cong S_3 \times C_2$

Prove that $D_{12}\cong S_3 \times C_2$. I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how ...
5
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2answers
74 views

Help to prove that $ U_{p} $ is a cyclic group.

As part of my study of Abstract Algebra, I’m trying to prove that $ U_{p} $ is cyclic for $ p $ a prime number. It’s a classical result, but I’m trying to prove it following four steps stated as ...
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2answers
34 views

Group Theory and Lagrange's Theorem: coprime subgroups. [duplicate]

Let $G_1$ and $G_2$ be finite groups, and let $K≤G_1 \times G_2$. Let $H_1 = \{ g \in G_1 : (g,e) \in K\}$ and $H_2 = \{g \in G_2 : (e,g) \in K\}$ and suppose $|G_1|$ and $|G_2|$ are coprime. Then ...
0
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1answer
47 views

Show that this group is nilpotent.

Let $G$ be a finite solvable group whose order is divisible by at least three distinct primes. If every Hall $p'$-subgroup of $G$ is nilpotent, show that $G$ is nilpotent. I feel like the best ...
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0answers
34 views

Need an example

Let $p$ be a prime number. I need an example of finite group $G$ generated by the elements of order $p^n$ ($n\in \mathbb N$) , which contains a normal subgroup $H$ that is not generated by the ...
3
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1answer
76 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
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1answer
32 views

If $G$ is finite group that supersoluble then $G$ satisfy the maximal permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
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0answers
22 views

can say every group that satisfy in maximal permutizer condition then satisfy then permutizer condition

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
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1answer
24 views

How to find subgroup centralizer?

Having found that a group G has a normal Sylow 2-subgroup P, how do I find $C_P(g_i)$, where $g_i$ is a conjugacy class representative? I have the character table, and have previously found ...
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1answer
23 views

What is the group $C_2^4$?

I'm trying to do a problem which asks me to show that a certain group is isomorphic to $C_2^4$. What is this group?
2
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1answer
33 views

Subgroups of every order dividing the order of the group imply the group is abelian?

Let $G$ be a finite group, denote $|G|=n$. I know about Cauchy theorem which states that if for a prime $p$: $p|n$ then there is $H\leq G$ with $|H|=p$. I also know that an abelian group $G$ have a ...
2
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2answers
31 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
3
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0answers
61 views

Normal subgroup of General linear group

What is the list of all normal subgroups of general linear group $GL_n(q)$? (n*n invertible matrix on finite field with $q$ elements) It is well known $SL_n(q)$ and subgroups of $Z(GL_n(q))$ are ...
0
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1answer
45 views

Coercing elements into a set in MAGMA

Suppose I had a permutation group $G$ for example, that corresponded to the intersection of some elements of a list $H$ where the elements are groups. Then, if I wanted to create a set or some ...
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2answers
16 views

Two subgroups whose orders are greater than square root of the group order have no trivial intersection

Two subgroups whose orders are greater than square root of the group order have no trivial intersection I cannot come up with a critical idea.
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1answer
62 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
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0answers
40 views

Fundamental Domain of ${\mathbb Z}^2$ to ${\mathbb R}^2$

Find a fundamental domain for the action of $\mathbb Z^2$ on $\mathbb R^2$ by translation A fundamental domain is the nodes $(0,0),(1,0),(0,1)$ and the edges which connect them Is there a better way ...
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0answers
32 views

subsets of $\mathbb{Z}_2^{p}$ up to permutation equivalence

Let $\mathbb{Z}_2:= \mathbb{Z}/2\mathbb{Z}=\{0,1\}$. Let $p$ be a prime integer. We use $$\mathbb{Z}_2^{p}:= \mathbb{Z}_2\times \mathbb{Z}_2 \cdots \times\mathbb{Z}_2\qquad (p-times).$$ i.e., each ...
1
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1answer
55 views

Non-trivial group homomorphism from an infinite group to a finite group

Let $G$ be a topological group the underlying set of which is infinite (e.g., $(\mathbb{R}\,;+)$ or $(\mathbb{Z}\,;+)$), and let $H$ be a topological group the underlying set of which is finite (e.g., ...
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1answer
23 views

Uniquely $p$-divisible group-Reference Request.

Define a group $G$ to be uniquely $p$-divisible if for all $x\in G$, there is a unique $y\in G$ such that $x=y^p$. Can someone kindly provide references where this class of groups is studied? Of ...
0
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1answer
24 views

If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

If $G$ is a finite group such that their is a normal subgroup $H$ of $G$ such that $G/H \cong V_4$ , where $V_4$ is the Klein four group , then is it true that $G$ can be written as a union of three ...
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42 views

An Abstract Characterization of $S_5$ using involutions and their centralizers

This is essentially an exercise from Jacobson's Basic Algebra I. (p.83, ex.10) I've managed to solve all the other part of the proof, except (vi) and (x). I've been thinking about this all day, but ...
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1answer
38 views

question on lemma in Bushnell and Henniart, irreducible components of a particular induced representation

I have a question on a lemma that appears in the book "The Local Langlands Conjecture for GL(2)" by Bushnell and Henniart. The setting is as follows: we let $G = GL_2(k)$ where $k$ denotes a finite ...
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1answer
20 views

Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
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34 views

Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
5
votes
2answers
61 views

If $|G|=p^n$, then $p^2 \le |G : G^\prime|$. [closed]

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
3
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1answer
84 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
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1answer
56 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
3
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1answer
51 views

If two subsets $S,T\subseteq G$ have sum of cardinalities greater than $|G|$, then $S+T=G$ [duplicate]

Let $S$ and $T$ are two subset of a finite group $(G,+)$ so that $|S|+|T|>|G|$, then Prove that $S+T=G$, where $S+T=\{s+t:s\in S ,t\in T\}$ My effort: It is clear that $S+T\subseteq G$ as ...
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$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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1answer
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Subgroup of a cyclic finite group [closed]

Let $G$ be a cyclic group of order $n$ and let $m$ be a positive integer dividing $n$. Show that $G$ contains one and only subgroup of order $m$. I have started the proof by stating the smallest ...
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2answers
34 views

Sylow counting - classifying groups of order 15

let $G$ be a group of order $15$. this is the argument I was given: We have $15 = 3\times 5$ so we start with $p = 5$ We have by Sylow's theorem that $N_5 = 1 \mod 5$ so $N_5 = 1$ or $N_5 \geq 6$. ...
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0answers
57 views

Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
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2answers
35 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
5
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1answer
57 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
2
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1answer
44 views

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$