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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8
votes
1answer
93 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
2
votes
2answers
310 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
4answers
80 views

Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism.

I'm stuck on this proof. I need to prove: Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism. And this is the ...
2
votes
1answer
58 views

About finite $p$-group finitely generated.

Let $p$ a prime number. Let $G$ be a $p$-group finitely generated, say, $G = \left<x_1, x_2, \ldots, x_m \right>$. Denote by $\gamma_i(G)$ the $ith$ term of the lower central series of $G$. ...
1
vote
2answers
70 views

Question about Sylow's theorems and a particular group of order 60

I have a finite group $G$ with the following data: Its order |$G$| is 60, it has exactly 6 Sylow-5-subgroups $P_i$$\ $ (i=1,...,6) and |$N_G(P_i)$|=10 $\forall$ i. I have the following questions: ...
2
votes
3answers
83 views

Show that A is a field which has $9$ elements

For $A=\left\{\left( {\begin{array}{*{20}{c}} a&b\\ { - b}&a \end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
5
votes
1answer
115 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
5
votes
1answer
200 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
3
votes
1answer
197 views

Orders of centralizers $C_G(g)$ in a group of order 60?

Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$? By considering ...
1
vote
3answers
117 views

$\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$.

The question is this: Prove that $\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$. I tried to construct a mapping from $f\colon\mathbb Z_n\to \mathbb Z_n$ and $f([k])=[ka]$ where ...
1
vote
1answer
341 views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by ...
5
votes
1answer
798 views

Why is $S_5$ generated by any combination of a transposition and a 5-cycle?

Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
0
votes
1answer
115 views

Right cosets of $K=\{1,17\}$ in $U_{32}$

How can we list the distinct right cosets of $K=\{1,17\}$ in $U_{32}$, the set of positive integers relatively prime to $32$?
2
votes
3answers
121 views

Show that for G be a abelian group, $g \in G, g^m=1$, where $m=|G|$

I am pretty sure there is a name for the following theorem, but unfortunately, I don't known it. Theorem. Let $G$ be a abelian group with $m=|G|$, than for any $g \in G, g^m=1$. Currently I don't ...
1
vote
2answers
249 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
4
votes
2answers
60 views

Existence of element of order $l$ dividing the order of the group

In this post: Order of kernel of a homomorphism , someone say that since $l$ divides $m$, we can say that there exists some element $x \in \ker (\varphi)$ such that $o(x)=l$. But why is it true? I ...
0
votes
1answer
65 views

How I can prove tht $T$ is isomorphic to a finite set of natural numbers?

Let $T$ be a finite abelian group. We can consider $T$ the as group $ℤ/nℤ$ or $ℤ/qℤ×ℤ/mℤ$. My question is: How I can prove tht $T$ is in bijection with a finite set of natural numbers? That is, I ...
0
votes
2answers
61 views

“Lifting the centralizer”

Let $G$ be a finite group, $T\le G$ and $N\unlhd G$ with $(|N|,|T|)=1$. Clearly $T$ acts by conjugation on $G$ and $N$ is a $T$-invariant subgroup; for this reason $T$ induces naturally an action on ...
5
votes
2answers
228 views

Upper bound for the sum of the orders in a finite group

This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution. Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
4
votes
2answers
1k views

A group of order 30 has a normal 5-Sylow subgroup.

There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me. Lemma Let G be a group of order 30. Then the 5-Sylow subgroup of G is normal. ...
1
vote
2answers
711 views

Orders of elements in cyclic groups

I think I'm a bit confused about the order of elements in cyclic groups. If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, ...
0
votes
2answers
93 views

Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
4
votes
2answers
104 views

Frattini Subgroup of p-Groups

Letting $P$ be a $p$-group and $\Phi(P)$ be the Frattini subgroup of $P$ (the intersection of all maximal subgroups), the challenge is "Prove that $P/N$ is elementary abelian implies $\Phi(P)≤N$" ...
2
votes
3answers
55 views

Computing $\langle (13746) \rangle$ in $S_7$.

How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
5
votes
5answers
112 views

How to prove $x \in H$

How to prove that Let H be a normal subgroup of a finite group G. If $\gcd(|x|, |G/H|)$ = 1, show that $x \in H$.
4
votes
3answers
424 views

Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$

Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$. Can we prove that no element has order bigger than $2^{k-2}$ ? My solution (not really a solution) : Since $n=2^k$, I ...
3
votes
1answer
106 views

Conjugation on subgroups of $A_4$ faithful?

Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$ I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
2
votes
0answers
54 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
0
votes
1answer
59 views

Finding a subgroup of Multiplicative group $\mathbb Z_{32}$

I am backing on some basic points about the multiplicative groups, like $\mathbb Z_{32}$ ,to review and I am really in a bad confusion to write the elements of a subgroup of it. For example, I want to ...
-1
votes
1answer
49 views

What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?

I know that the order of every $T∈ℤ/nℤ$ divides the size of the group $n$. My question is: What is the order of an element in $ℤ/2mℤ×ℤ/2ℤ$?
1
vote
2answers
166 views

Group theory problem to be solved?

Let $G = S_n$, the symmetric group of order $n$, acting as permutations on the set $\{1,2,\dots,n\}$. Let $H = \{\sigma \in G \mid n \cdot \sigma = n\}$. (i) Prove that $H$ is isomorphic to ...
5
votes
2answers
171 views

Prove that $S_4$ has no subgroup isomorphic to $Q_8$.

The question is to prove that $S_4$ has no subgroup isomorphic to $Q_8$. Here is an answer. But what "then $H$ also contains all products of two 2-cycles" means in that answer? Thanks.
4
votes
1answer
88 views

Groups and Lagrange theory

There are two subgroups $H_1$, $H_2$ of $G$, if $H_1\neq H_2$ then $\gcd(|H_1|,|H_2|)=1$. Prove that the order of $G$ is a prime number and the group is cyclic. I know from Lagrange that the order ...
-1
votes
2answers
112 views

Cyclic Group of order $8$

Let $G=(a)$ be a cyclic group of order $8$ and let $H=(a^4)$ be its subgroup of order $2.$ Find the coset representation of $G$ by $H$.
0
votes
1answer
72 views

Prove if $g$ is an element of order $d$ and $d$ divides $n$ then $gn = 1$. [duplicate]

Prove if $g$ is an element of order $d$ and $d$ divides $n$, then $gn = 1$.
5
votes
3answers
1k views

Show that $N$ is a normal subgroup in $G$ when $N$ is the intersection of normal subgroups in $G$

QUESTION : Let $G$ be a group, let $X$ be a set, and let $H$ be a subgroup of $G$. Let $$N = \bigcap_{g\in G} gHg^{-1}$$ Show that $N$ is a normal subgroup of $G$ conitained in $H$. MY ATTEMPT: I ...
1
vote
2answers
2k views

Concatenation of 2 finite Automata

I have some problems understanding the algorithm of concatenation of two NFAs. For example: How to concatenate A1 and A2? A1: ...
6
votes
3answers
331 views

Does $G\times K\cong H\times K$ imply $G\cong H$?

Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?
3
votes
1answer
47 views

Maximal Subgroups Containing given Element

Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?
1
vote
3answers
337 views

Direct product of finite cyclic groups of coprime orders [duplicate]

The Question is this: How many generators are there of the group $G\times H$, if $G$ and $H$ are cyclic groups of order $m$ and $n$, which are coprime? Let's say that $G$ is generated by $g$, and ...
2
votes
2answers
67 views

Verifying homomorphism $S_3 \to \langle \phi \rangle$

Let $$ \phi : \begin{array}{c} 1 \mapsto 2 \\ 2 \mapsto 1 \\ 3 \mapsto 3 \\ \end{array} \qquad \text{and} \qquad \psi : \begin{array}{c} 1 \mapsto 2 \\ 2 \mapsto 3 \\ 3 \mapsto 1 \\ \end{array}. $$ ...
0
votes
1answer
226 views

Prove that [GxH : AxB]=[G:A][H:B] when A < G and B < H

The original question is that: If A is subgroup of group G and B is a subgroup of group H, then express [GxH : AxB] in terms of [G:A] and [H:B] and prove the result is correct! Then I first prove ...
2
votes
2answers
195 views

Sylow Theorems Question

Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$-subgroup of $G$ and is cyclic. Prove that $N(H)=C(H)$.
-2
votes
2answers
111 views

Elements of a given order in finite cyclic groups

List all elements of order $4$ in $\mathbb{Z}_8=\mathbb{Z}/8\mathbb{Z}$. Also list all the elements of order $6$ in $\mathbb{Z}_{72}=\mathbb{Z}/72\mathbb{Z}$.
0
votes
2answers
147 views

Product of disjoint cycles question.

Consider the following permutations $x$ and $y$ in $S_6$: $x=(1 \, 3 \, 5)(2 \, 4)$ and $y=(2 \, 3 \, 4 \, 5)$ Express $xy$ as a product of disjoint cycles. My attempt: I first got $xy = (3 \, 5 \, ...
11
votes
0answers
255 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 ...
3
votes
5answers
430 views

Cayley's Theorem question: examples of groups which aren't symmetric groups.

Basically, Cayley's Theorem says that every finite group, say $G$, is isomorphic to a subgroup of the group $S_G$ of all permutations of $G$. My question: why is there the word "subgroup of"? If we ...
-1
votes
1answer
160 views

Isomorphisms between symmetric, dihedral and cyclic groups

What examples are there of isomorphisms between the groups $S_n,\, D_n, \, \mathbb{Z}_n$? Thank you.
3
votes
1answer
229 views

Inverse Scalar Multiplication of a point over elliptic curve

I was implementing point arithmetic operation, and was exploring the properties of point arithmetic, and I am unable to conclude whether $$ k^{-1}(kP) = P $$ where P is a point over elliptic curve $ ...
1
vote
2answers
2k views

Cayley's Theorem - Discourse on Fraleigh's proof or Alternative Easier proof [duplicate]

I have hard time understanding Fraleigh's proof. Can someone either explain Fraleigh's or provide an alternative, perhaps easier proof? Thanks so much.