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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3answers
143 views

If $G$ is a finite group and $H \subset G$ is closed, must $H$ be a subgroup?

Supposed theorem (from online notes): If $(G,*)$ is a finite group and $H\subset G$, $H$ is non-empty and $H$ is closed under $*$ then $(H,*)$ is a group. The proof given is a real mess, but, after ...
0
votes
1answer
42 views

finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6

If I have a finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6, is there anything special about G that we can infer? Would the order of $G$ be 30?
4
votes
1answer
64 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
3
votes
1answer
167 views

What exactly is a p'-group?

In the context of finite group theory I understand that $p$-group is a group whose order is a power of $p$ ($p$ a prime number) but I am unclear on the exact definition of a $p'$-group. This ...
0
votes
0answers
41 views

Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
2
votes
2answers
27 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
0
votes
1answer
54 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that $12p=2^2\...
3
votes
1answer
53 views

Is the following equation equivalent to the 2-cocycle condition?

Given a finite abelian group $G$, I'm looking for functions $\rho:G \times G \to U(1)$ such that 1) $~~~~~\rho(g,e) = 1 = \rho(e,g)$, where $e\in G$ is the unit element and such that 2) $~~~~\...
2
votes
1answer
48 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow $p$-...
2
votes
1answer
55 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
0
votes
0answers
18 views

Vertex-transitivity of the automorphism group of a digraph

I am trying to understand the theorem 3 of Cycles in graphs and groups by Kantor. Theorem $3$ If $G$ is a vertex-transitive group of automorphisms of a digraph $\Gamma$ with outdegree $d \ge 1$, ...
0
votes
0answers
59 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
7
votes
1answer
65 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
0
votes
0answers
45 views

I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
0
votes
0answers
15 views

Intuition of a theorem in Abstract groups

A theorem in Abstract groups Let $N \triangleleft G$ Then, $1\cdot $ If $H \leq G$ with $N \leq then H/N \leq G/N.$ Morever, if $N \leq K \leq G with K/N =H/N$ then $K=N$ ...
0
votes
1answer
38 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
0
votes
2answers
28 views

Map of an element in a group to the conjugation by g

Let G be a group and suppose $g \in G$. $\varphi:G\rightarrow Aut\left ( G \right )$ $g \mapsto i_{g}$ is a Homomorphism with image $Inn\left ( G \right )$ where $Inn\left ( G \right )=\left \{ i_{...
3
votes
1answer
47 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
0
votes
1answer
37 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
0
votes
2answers
34 views

Finite abelian groups isomorphic?

I know cyclic groups of the same order are always isomorphic, but as far as I'm aware finite abelian groups aren't necessarily cyclic. So is this statement true or false, and why?
1
vote
0answers
21 views

units of the quotient ring of the integers over a prime power $[{\Bbb Z}/P^e\Bbb Z]^*$ is cyclic multiplicative group

I am studying Algebra as an extra curricular research project and in the reading I was assigned, the author somewhat offhandedly mentions that the units of ${\Bbb Z}/P^e\Bbb Z$, which is to say $$({\...
0
votes
0answers
32 views

Are there $n$ nilpotent groups of order $n$ for some $n>1$?

Denote $nil(n)$ to be the number of nilpotent groups of order $n$. I checked the numbers $1<n\le 10^7$ , such that $n$ is neither divisble by $2^{11}$ nor a seventh power of a prime. None of ...
1
vote
0answers
23 views

Prove a function to be an automorphism of a $p$-group $G$.

Let $M$ be a maximal subgroup of a $p$-group $G$. For fixed $g\in G\backslash M$ and $z\in Z(G)\cap M$ of order $p$, the map \begin{align*} \alpha : G&\longrightarrow G\\ mg^i &\...
0
votes
1answer
47 views

infinite cyclic group is isomorphic to the group of integers under addition.

Theorem: Every finite cyclic group is Isomorphic to $\left ( \mathbb{Z},+ \right )$ Proof: We show first that the map $\phi$ is a homomorphism. Then, show that $\phi$ is a bijection. Showing $\...
0
votes
1answer
77 views

First Isomorphism proof

Theorem: First Isomorphism Theorem Let G and H be groups $\varphi :G\rightarrow H$ a Homomorphism Then, $G/\ker(\varphi) \cong \varphi(G)$ via the Isomorphism $$\Psi:G/\ker(\varphi)\...
0
votes
1answer
18 views

Deducing information about the normal subgroups of a finite group $G$ from its finite cyclic homorphic image?

The following example is taken from the book "Contemporary Abstract Algebra" by Joseph A. Gallian, seventh edition, page#210. If $G$ is a group of order $60$ and $G$ has a homomorphic image of order $...
0
votes
1answer
33 views

Estimating the cardinality of the union of the conjugates of a proper subgroup.

If $G$ is a finite group and $H$ is a nontrivial subgroup of $G$ such that $H^{a}\cap H=\{e\}$ for all $a\in G-H$, where $H^{a}=aHa^{-1}$ and $e$ is the neutral element of $G$, show that $$\left\lvert\...
0
votes
1answer
17 views

Generating set - inconsistency?

In my lecture notes $‹S›$ is defined as follows: Then later there is this: But surely this is exactly what $‹s,t›$ is? Directly from the Proposition, with $S=${$s,t$}, $H=${$s^jt^k$} with $s,t∈S$...
0
votes
1answer
51 views

Number of groups of order $512$ with exponent $2,4,8,16,…$

I want to determine the number of groups of order $512$ with exponent $2,4,8,16,32,64,128,256$ and $512$ The first $500,000$ groups in GAP give the following result : ...
0
votes
1answer
20 views

Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
1
vote
0answers
27 views

Is $N_{A_7}(H) = H$, with the following $H$?

I am following a proof in which I have a subgroup of $S_7$ defined by $H := \langle (2, 3, 4)(5, 6, 7) , (2, 7, 6, 3)(4, 5) \rangle$ The book implicitly uses that $N_{A_7}(H) = H$ (the normalizer ...
2
votes
1answer
50 views

About conjugating a $7$-cycle in a subgroup of $S_7$

Following a proof in which I have a transtive group $G$ of order $168$ , which is a subgroup of $S_7$ (I am trying to characterize it, I cannot use well know facts such as it is always isomorphic to $...
0
votes
0answers
28 views

Left coset of H in $A_{4}$

This problem arises in problem 1 of Chapter 7 of Contemporary abstract algebra by Joseph Gallian. This problem requires the use of Table 5.1 on page 104 of Chapter 5 and I am unable to replicate the ...
0
votes
1answer
21 views

Subset of stabiliser is a group

Definition: Let G be a permutation group of a finite set $\Omega.$ Let $\alpha \in \Omega.$ The Stabiliser of the point element $\alpha$ in the group G is the set $G_{\alpha}=\left \{ ...
1
vote
1answer
19 views

A $p$-subgroup of a finite group is either a Sylow $p$-subgroup or properly contained in a Sylow $p$-subgroup of its normalizer

This exercice is from Aschbacher's book "Finite group theory". I am not asking for a complete solution, just for a hint. Here is a partial solution, when the ambient group $G$ is a $p$-group. If $X$ ...
0
votes
2answers
41 views

Does $S_6$ have an abelian sylow $2$ subgroup.

How do I check if $S_6$ has an abelian sylow 2 subgroup. Order of any sylow 2 subgroup is $16$ and by sylows theorem it has $45$ sylow 2-subgroups, but how to check whether it has any abelian sylow 2-...
1
vote
1answer
43 views

Proof to an observation of stabilisers and orbits

Observation: If $\alpha^{g}=\beta$ then $G_{\beta}=g^{-1}G_{\alpha}g$ Just to get the notation out of the way: $G_{\beta}= g^{-1}G_{\alpha}g$ is the stabiliser of a point element $\beta$ in a ...
2
votes
1answer
28 views

Example of a group in which intersection of all non-normal subgroups is non-trivial

What are some examples of a group in which intersection of all non-normal subgroups is non-trivial. Do I consider abelian or Hamiltonian groups an example for this as they have only normal subgroups, ...
0
votes
0answers
23 views

Prove that all the conjugates of a proper subgroup cannot cover whole group $G$ [duplicate]

Let $G$ be a finite group and $H\le G$. Then how do I prove that $G$ cannot be written as $\cup\ xHx^{-1}$ for $x\in G$
3
votes
2answers
41 views

An automorphism that has no fixed points except for the identity and is its own inverse implies commutativity

Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ of $...
1
vote
0answers
17 views

Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ a_1\...
0
votes
0answers
33 views

Reference request: planar Cayley graphs

In 1896, Maschke classified all finite groups that admit a planar Cayley graph. The paper is here: http://www.jstor.org/stable/pdf/2369680.pdf I've been trying to digest this paper, but I'm finding ...
0
votes
1answer
59 views

Explanation to Fermat's little theorem proof

Fermat's little theorem $\forall a \in \mathbb{Z}$ and every prime p. Then, $a^{p}\equiv a\pmod p$ $a=pm+r $ $\forall 0 \leq r<p$ Proof for $r\not\equiv 0:$ Then, $\forall r \in \bar{U}\...
3
votes
1answer
49 views

Groups which can not occur as automorphism group of a group

Consider the following natural question: Given a finite group $H$, does there exists a finite group $G$ such that $\mathrm{Aut}(G)\cong H$? In short, does any finite group occurs as the ...
1
vote
2answers
31 views

Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
0
votes
1answer
40 views

Show that $|a^{k}|=|a^{n-k}|$

Let G be a group and let $a$ be an element of G of order $n$. For each Integer $k$ between $1$ and $n$, show that $\left | a^{k} \right |=\left | a^{n-k} \right |$ My attempt is as follows: $\left |...
0
votes
0answers
21 views

Nontrivial example of t-groups.

A t-group is one in which every subnormal subgroup is normal. Now obviously all abelian or Hamiltonian groups are t-groups as all subgroups in them are normal. What would be an example of a t-...
2
votes
1answer
27 views

If $A,B$ is nilpotent then $G$ is solvable?

let $G$ be a finite group and let $A,B$ be two proper nilpotent subgroup. If $$AB=G$$ then is it true that $G$ is necessarily solvable ?
0
votes
1answer
32 views

How to find a normal subgroup in the $p-$group of order $p^5$ other than center?

Let $G$ be a non-abelian $p$-group of the order $|G|=p^5$ such that Frattini Subgroup $\Phi(G)$, Commutator Subgroup $G'$ and Center $Z(G)$ of $G$ are equal and Rank($G$)=$3$, Exponent($G$)=$p$ (there ...
0
votes
0answers
25 views

Element of infinite order and all generator of subgroups

Suppose that a has infinite order Find all generators of subgroup $\left \langle a^{3} \right \rangle$ Now, since a has infinite order then so does $\left ( a^{3} \right )^{n} $for if a has finite ...