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
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1answer
36 views

A specific group lacking Følner sequences

How does one go about proving that the free group $<a,b,a^{-1},b^{-1}>$ lacks any Følner sequence?
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vote
1answer
53 views

If G is finite p-group and H is a subgroup, show that there is a composition series that contains H.

Let $p$ be a prime number, suppose $G$ is a finite p-group and let $H$ be a subgroup of $G$. Show that there is a composition series that contains $H$. I have already shown that if $G$ if a finite ...
1
vote
0answers
63 views

Is this subset of group, a subgroup of it? [duplicate]

Suppose that $G$ is a finite group, $n = |G|$, and $X$ be a non-empty subset of $G$. Is it true that $$ H := \{ x_1 x_2 \dots x_n | x_i \in X \} $$ is a subgroub of $G$? Edit: Since $n=|G|$ so $H$ ...
2
votes
2answers
52 views

A question on finite group

Let $G$ be a finite group and $p$ be the smallest prime divisor of $|G|$ , let $x \in G$ be such that $o(x)=p$ , and suppose for some $h\in G $ , $hxh^{-1}=x^{10}$ , then is it true that $p=3$ ?
0
votes
0answers
33 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 ...
1
vote
0answers
22 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
2
votes
0answers
22 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
1
vote
1answer
74 views

showing that $G$ is nilpotent.

suppose $G$ is a finite solvable group,then $G$ is nilpotent if and only if all Hall subgroups of $G$ which its indices are power of a prime number are normal. suppose $G$ is a solvable finite group ...
2
votes
1answer
50 views

showing that $G$ is not solvable.

suppose $G$ is a finite group and $1\neq a \in G$ ,$1\neq b \in G$ and $O(a)$ ,$O(b)$ ,$O(ab)$ every two of them are relatively prime ,then $G$ is not solvable. my Idea:I suppose $G$ is solvable and ...
3
votes
0answers
66 views

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 ...
0
votes
1answer
72 views

Question about Z12≃Z4⊕Z3 generators and order

I apologize in advance for my messy language and questions; I've only been studying group theory for a month and thus these concepts aren't clearly locked in yet; hence my questions. :) Take ...
1
vote
1answer
22 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
1
vote
2answers
117 views

Any group which is of prime order is a cyclic group

I don't know how to prove this: Any group which is of prime order is a cyclic group. What fact should I use to prove this?
1
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1answer
81 views

proof of a useful counting result in group theory

Let G be a finite group, H a subgroup of G satisfying |G| |̸| [G : H]!. Prove there exists a normal subgroup N of G satisfying 1 < N ⊂ H. maybe the General Cayley's Theorem works. I am not sure. ...
4
votes
3answers
52 views

if $k$ is a positive integer and $G$ a finite group such that $G=\{x^k:x\in G\}$ , then is it true that g.c.d.$(|G|,k)=1$ ?

If $G$ is a finite group of order $n$ and $k$ is a positive integer such that g.c.d.$(n,k)=1$ , then I know that $G=\{x^k:x\in G\}$ ; is the converse true ? that is if $k$ is a positive integer and ...
0
votes
2answers
28 views

$S$ nonempty finite subset of group $G$ for which $SS=S$. $S$ is subgroup.

Let $S\subset G$, $S$ finite and nonempty, $G$ group. Suppose additionally that $$SS=\{s_1 s_2: s_1\in S, s_2 \in S\}=S.$$ How can I prove that $S$ is a subgroup of $G$? Does this hold for $S$ ...
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vote
2answers
81 views

Is $ G \cong G/N \times N$?

If G is a finite group and N is a normal subgroup in G , then can we say G $\cong$ G/N $\times$ N always? Is it true for like normal nilpotent or normal solvable or any such special classes. I ...
-1
votes
1answer
45 views

Automorphism Tower for $\Bbb Z_4$, $\Bbb Z_3$, $\Bbb Z_2 \times \Bbb Z_2$

Let $G$ be a group. Now consider $\operatorname{Aut} G$ then $\operatorname{Aut} (\operatorname{Aut} G) = \operatorname{Aut}(G^2)$, then $\operatorname{Aut}(\operatorname{Aut}(\operatorname{Aut} ...
3
votes
2answers
75 views

$G$ is product of its center and commutator

Let $G$ be a group s.t. $G=Z(G)G'$. When $G$ is abelian or perfect, the above equality is trivially true. We can also construct an example like $\mathbb{Z}_3 \times A_5$ i.e. a product of abelian and ...
7
votes
4answers
133 views

How many $A_5$ are there inside $A_6$?

I am reading a paper that says: There are $12$ versions of $A_5$ in $A_6$: $1) $ the permutations that leave one thing unmoved. $2)$ the permutations of the six pairs of antipodal ...
0
votes
0answers
9 views

How to prove the orthogonal relation between two different Young operators?

In the group theory, Young diagram, Young tableau and Young operator are all valuable concepts. For any given two Young diagrams, how can we prove that the Young operators($Y$ and $Y^{'}$) ...
2
votes
1answer
19 views

Question about classifying semidirect product

I have in some notes, this statement: Given $C_3\ltimes C_7$ we know that for $a\in C_3$ and $b\in C_7$, and some $k$: $$aba^{-1}=b^k$$ $$k^3\equiv 1(7)$$ The reason given is that $a^3=1$. ...
4
votes
1answer
41 views

Finite abelian groups and subgroups.

Let $G$ be a finite abelian group of order $n=p_1^{a_1}\cdot \cdot \cdot p_k^{a_k}$ and $H$ a subgroup of $G$ of order $m=p_1^{b_1}\cdot \cdot \cdot p_k^{b_k}$. By Theorem 5 on page 161 of Dummit ...
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vote
2answers
101 views

Prove that $G$ has a subgroup isomorphic to $G/H$.

Let $G$ be a finite abelian group of order $n$ and let $H$ be a subgroup of $G$ of order $m$. Show that $G$ has a subgroup isomorphic to $G/H$. Here are my thoughts: Define $\mu_n := \{z \in ...
2
votes
1answer
39 views

Subgroup of a p-group in its center

Suppose $p$ is prime, $n\in\mathbb Z^+$, and $G$ is a group of order $p^n$. If $H$ is a subgroup of order $p$ and $ghg^{-1}\in H$ for all $g\in G$ and $h\in H$,I can't seem to show that $H \subseteq ...
1
vote
1answer
67 views

Cayley graphs of finite 2-generator groups

Let $G=\left<a,b\right>$ be a finite 2-generator group and $\Gamma$ its Cayley graph with respect to $\{a,b\}$. Is it true that $\Gamma\setminus\{e,f\}$ is connected for two arbitrarily chosen ...
2
votes
3answers
44 views

Prove that $C_{A_7}((567))\cong H \times A_4$

Let $H=(<567>)\subset{A_7}$. And let $C=C_{A_7}((567))$ denote the centralizer of $(567)$ in $A_7$. Prove that $C_{A_7}((567))\cong H \times A_4$ I'm fairly certain that I can use the First ...
2
votes
1answer
50 views

all Sylow subgroups of $GL_n(\mathbb{F}_q)$

Can you give some references to find all Sylow subgroups of $GL_n(\mathbb{F}_q)$? I know that upper triangular matrices with diagonal's 1 is a Sylow $p$-subgroup where $q=p^n$. But how about the other ...
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0answers
44 views

Help with proof for problem 14 chapter 5 Dummit and Foote.

For any group $G$ define the dual group of $G$ (denoted $\hat{G}$) to be the set of all homomorphisms from $G$ into the multiplicitive group of roots of unity in $\mathbb{C}$. Define a group operation ...
2
votes
1answer
23 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
5
votes
1answer
88 views

Show that a group of order $180$ is not simple.

What i deduced is that $n_5=1,6$ or $36$. We are done if $n_5=1$. If $n_5=36$ we $N_G(P)=P$ for any Sylow $5$-subgroup P as $|N_G(P)|=\frac{180}{36}=5$ and $P$ is abelian cyclic so by Burnside ...
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0answers
54 views

Generalization of a property of finite nilpotent groups

Let $G$ be a finite nilpotent group and $M$ be a maximal subgroup of $G$. If $H$ is a proper non-trivial subgroup of $G$ such that $H\not\leq M$, then we can show that $H\cap M$ is a maximal subgroup ...
0
votes
3answers
53 views

Why is this map well-defined?

Let $G$ a finite group and $H$ and $K$ two sub-groups of $G$. Why is the map \begin{array}{rcl} \Psi: G/({H\cap K}) & \longrightarrow &G/H \times G/K \\ g(H\cap K) & \longmapsto ...
0
votes
2answers
55 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
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vote
1answer
28 views

Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$

I am reposting it getting insufficient help from the previous post (Although I got some hint) Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), ...., ...
4
votes
2answers
114 views

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$?

Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2013}$? Clearly, we can assume the Structure Theorem for finite abelian groups. Edited Later: All I ...
0
votes
0answers
40 views

Show that $a\mapsto a^n$ is an isomorphism when $\gcd(|G|,n)=1$.

Let G be a finite abelian group and let n be a positive integer that is relatively prime to $|G|$. Show that the mapping $\phi:G\to G$ given by $a\mapsto a^n$ is an isomorphism. I solved homomorphism ...
0
votes
1answer
38 views

Action on $G$ by inner automorphism

I wonder something about an action of a group $A$ on a group $G$ by a automorphism; There are many nice result related with some restrictions such as when $(|A|,|G|)=1$ , $G$ is abelian or ...
3
votes
3answers
66 views

What group is $(\mathbb{Z}/24\mathbb{Z})^{*}$ isomorphic to

I want to determine which group $(\mathbb{Z}/24\mathbb{Z})^{*}$ is isomorphic to. $\mathbb{Z}/24\mathbb{Z}$ contains the 24 residue classes $z + 24\mathbb{Z}$ of the division mod 24. For brevity, I ...
0
votes
2answers
38 views

$|G|=p^3$, prove that $p$ divides |Z(G)|. [duplicate]

Suppose that a group $G$ has order $p^{3}$ where $p$ is prime. How would I prove that $p$ divides $|Z(G)|$?
2
votes
2answers
216 views

Are there any finite non-abelian group with one subgroup of each size ?

Let $G$ be a finite group with at most one subgroup of any size , then is it true that $G$ is cyclic ? I can prove that the answer is "yes" with the additional assumption of abelian ness on $G$ but ...
7
votes
2answers
61 views

Embedding of finite groups into general linear group

It's clear that for any field $\mathbb{F}$ any finite group $G$ can be embedded into $GL_{n}(\mathbb{F})$ for some $n$. My question is about one modification of this result. Let's fix positive ...
5
votes
1answer
84 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
2
votes
2answers
56 views

Every finite $p$-group is solvable

I know that in some version of Sylow's 1st theorem, it states that if $|G|=p^nm$ for some $n\geq 1$ and where $p$, a prime number, does not divide $m$, then every subgroup $H$ of $G$ of order ...
3
votes
0answers
43 views

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 ...
-3
votes
1answer
55 views

Normal subgroup contained in the center [closed]

If a normal subgroup $N$ of order $p$, $p$ prime, is contained in a group $G$ of order $p^n$, then $N$ is in the center of $G$.
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votes
2answers
45 views

Subgroup of prime order which is normal [closed]

If $|G| =pn$, with $p>n$, $p$ a prime and $ H$ is a subgroup of order $ p $ then $ H $ is normal in $ G $.
1
vote
1answer
41 views

Every $p$-subgroup is contained in one $p$-Sylow subgroup?

I am learning Sylow's theorems in my algebra course and I was reading questions posted before. One is the following: If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. ...
1
vote
1answer
35 views

Subgroups of Order $p^2$ in $\mathbb{Z}_p \oplus \mathbb{Z}_p$

Hello Mathematics Community. I am unsure about how to solve this problem involving the number of subgroups in an abelian group. How many subgroups of order $p^2$ does the abelian group $\mathbb{Z}_p ...
1
vote
1answer
40 views

The structure of maximal tori in finite simple groups

Let $\mathbf{G}$ be a linear algebraic group over an algebraic closed field of characteristic $p$ and $F$ a proper frobenius map on it with fixed point group $\mathbf{G}^f=G$ such that ...