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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2
votes
2answers
28 views

Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
2
votes
1answer
29 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
2
votes
4answers
100 views

If G is a group not cyclic then its order can be:

If G is a group not cyclic then its order can be: a)15 b)35 c)77 d)120 e)2011 Well, i know that if G is not cyclic then it is not isomorphic to Zn, but i think it does not help much. Any ...
-1
votes
1answer
22 views

Equivalent permutations

Is this possible, when two groups are isomorphic then their group generators are equivalent? Moreover, by definition if two permutations $u,v$ are equivalent then either $u$ can be obtained from $v$ ...
1
vote
2answers
42 views

Let $p$ be a prime number; and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G:G]$

I have already figured out that $|Z(G)| = p$, and that $G'=[G:G] \lhd G$. Also, $|G'| = p$ or $p²$ I suppose I'd have to prove that $G' \subset Z(G)$, but I've been trying and I have no idea how. ...
0
votes
1answer
41 views

What are all the subgroups of $\Bbb{Z}/10\Bbb{Z}$?

I thought they would be $\Bbb{Z}/n\Bbb{Z}$, where $1 \leq n \leq 10 $. What is wrong in that ? And, how is $\Bbb{Z}/n\Bbb{Z}$ a cyclic group. Is it's generator the element 1 ?
0
votes
0answers
13 views

scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
1
vote
1answer
34 views

Topics in Group Theory [on hold]

Actually I am going to do summer project (2 month project) on Group Theory so could you please let me know what are the topics in Group Theory on which I can do project. Please keep in mind that I am ...
1
vote
2answers
42 views

Normal subgroups and index problem

Let $G$ be a finite group and $H$ and $K$ subgroups such that $H \lhd G$ or $K \lhd G$. If $gcd(|K|;|G:H|)=1$, show that $K \subset H$. I think I could prove it for the case $H \lhd G$ Let $k \in ...
1
vote
1answer
30 views

How does GAP understand $SL_2(\mathbb{F}_3)$?

When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP ...
1
vote
1answer
37 views

Prove the existence of order 4 subgroups of order 8 groups

I am participating in an Introductory course in groups and I have the following question: Let $G$ be a finite group of order $8$. Prove that $G$ has a subgroup of order $4$ and a subgroup of order ...
3
votes
2answers
53 views

Finite subgroup of $\mathbb C^{\times}$

I was trying to show that every finite subgroup of $\mathbb C^{\times}$ is equal to $G_n$ (the nth roots of unity) for some $n \in \mathbb N$ without invoking Lagrange's theorem, I got stuck at one ...
0
votes
1answer
21 views

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p = q$.

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p=q$. First of all, I'm a bit confused about the meaning of '... are conjugate'. Does this mean that ...
0
votes
0answers
26 views

Let $ H $ be a subgroup of $ G $ auch that $ | G:H | $ is a $ \pi $-number [on hold]

Let $ H $ be a subgroup of $ G $ such that $ | G:H | $ is a $ \pi $-number. If there is a nilpotent subgroup $ K $ of $ G $ such that $ G = HK $. let $ K = K_{\pi^{\prime}}K_{\pi} $, where $ ...
0
votes
1answer
44 views

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 ...
1
vote
0answers
23 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
votes
2answers
42 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
votes
2answers
83 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? ...
0
votes
1answer
41 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
1
vote
1answer
72 views

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

For example for the group $SL_2(\mathbb{F}_3)$ I get the following, ...
1
vote
1answer
19 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, ...
7
votes
0answers
83 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 ...
0
votes
1answer
37 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) ...
1
vote
1answer
42 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 ...
0
votes
2answers
31 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 ...
0
votes
1answer
43 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 $g \in G$ ...
1
vote
0answers
40 views

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ [closed]

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. I want to know which one of the following groups is capable? $(\mathbb{Z}_2\times D_8)\rtimes \mathbb{Z}_2$, ...
3
votes
2answers
51 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 ...
0
votes
0answers
19 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) ...
7
votes
3answers
49 views

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
votes
2answers
61 views

Help to prove that a group is cyclic

As part of my study of Abstract Algebra I'm trying to prove that $U_p$ si cyclic for $p$ a prime number. It's a classical result, but I'm trying to prove it following 4 steps stated as problems in my ...
-1
votes
0answers
30 views

Cyclic subgroups of $\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$

$\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$ has 24 elements of order 10. Why each cyclic subgroup of order 10 has four elements of order 10 ?
2
votes
2answers
32 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
votes
1answer
43 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 ...
0
votes
0answers
33 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 ...
-1
votes
0answers
23 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$

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 ...
3
votes
1answer
61 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 ...
2
votes
1answer
30 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 ...
1
vote
0answers
20 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 ...
0
votes
1answer
18 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 ...
0
votes
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
votes
1answer
32 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
votes
2answers
30 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
votes
0answers
52 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
votes
1answer
21 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 ...
0
votes
2answers
14 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.
1
vote
1answer
47 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
0
votes
0answers
36 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 ...
0
votes
0answers
29 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
vote
1answer
41 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., ...