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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4
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2answers
87 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
52 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
110 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
373 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
92 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
51 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
54 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
46 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
161 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
161 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
83 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
106 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
66 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
903 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
1k 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: ...
1
vote
0answers
129 views

Order of kernel of a homomorphism [closed]

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
5
votes
3answers
281 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
46 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
221 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
64 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
177 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
183 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 $N(H)=C(H)$.
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votes
2answers
105 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
127 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 \, ...
10
votes
0answers
216 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
295 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
143 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.
2
votes
1answer
151 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.
5
votes
5answers
397 views

Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.

Let $G$ a group of order $6$. Prove that: i) $G$ contains 1 or 3 elements of order 2. ii) $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$. I haven´t covered Sylow groups and normal groups. ...
3
votes
5answers
259 views

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it? I came across the above problem and do not know how to get it? Can someone point me ...
6
votes
3answers
66 views

Is $PGL_2(q)$ isomorphic to $SL_2(q)$

Let $F_q$ denote the field of order $q$. Define: $GL_2(q)$ to be the group of invertible $2$ by $2$ matrices over $F_q$. $SL_2(q)$ to be its subgroup consisiting of invertible $2$ by $2$ matrices ...
8
votes
2answers
741 views

Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.

Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic. What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
0
votes
1answer
185 views

Right translation - left coset - orbits

We can remark that the left coset $gH$ of $g \in G$ relative to a subgroup $H$ of $G$ is the orbit of $g$ under the action of $H \subset G$ acting by right translation. What is that right ...
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vote
3answers
215 views

Group theory - left/right $H$-cosets and quotient sets $G/H$ and $G \setminus H$.

Let $G$ be a group and $H$ be a subgroup of $G$. The left $H$-cosets are the sets $gH, g \in G$. The set of left $H$-cosets is the quotient set $G/H$. The right $H$-cosets are the sets $Hg, g\in G$. ...
5
votes
1answer
596 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
2
votes
0answers
50 views

characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
1
vote
1answer
534 views

Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle

$\bf Claim:$ Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle. If $\alpha$ and $\beta$ are disjoint, they commute and thus the product ...
1
vote
5answers
192 views

Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$

Can you please help me in this question: Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$. Thanks a lot
1
vote
3answers
55 views

A question on finite $p$-groups. [duplicate]

Is true that if $G$ is a $p$-group finite, say, $\mid G \mid = p^d$, then $G$ is $d$-generated?
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vote
3answers
88 views

Normal Subgroups, index, divisible orders

Let $H$ be a normal subgroup of $G$ with index $k$ . Show that if $a \in G$ and $o(a)=n$, then the order of $aH$ in $G/H$ divides both $n$ and $k$ .
4
votes
3answers
138 views

Order of a permutation group

I'm playing with a permutation group, with generators $(1,2,3)(6,5,4)$ and $(2,5,7)(8,6,3)$, in cycle notation. After careful counting, I believe its order is 24. But I have no real method, except ...
1
vote
1answer
160 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
2
votes
1answer
142 views

Systems of coset representatives

I have the following question. Let $H\leq U\leq G$ be (not necessary finite) groups. Let $S$ is a System of Right coset represantatives of $U$ in $G$, i.e. $\bigcup_{s\in R} Us=G$ with $Us\cap ...
0
votes
1answer
27 views

$m'$-group being cyclic?

Given a group $G=\mathbb{Z}_m\rtimes\mathbb{Z}_n$ with $m,n$ coprime. Should every subgroup of $G$ that has order coprime to $m$ be cyclic?
5
votes
1answer
303 views

Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$

Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
3
votes
0answers
99 views

Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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vote
3answers
98 views

Some groups of order $40$

Is there some table on the web giving information about particular small groups, that would go up to order $40$ and that would give enough information so that one could be sure whether groups matching ...
2
votes
2answers
341 views

Order of kernel of a homomorphism.

Let $C_n$ denotes the cyclic group of order $n$ and let $\phi:C_{52}\rightarrow C_{52}$ be the homomorphism $\phi(x)=x^7$. What is the order of kernel of $\phi$? I know that $ker\phi=\left\{x/ ...
3
votes
1answer
194 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...