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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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 ...
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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 ...
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1answer
14 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 ...
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1answer
48 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 : ...
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1answer
18 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 $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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A $p$-subgroup of a finite group is either a Sylow $p$-subgroup or properly containted 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 $X$ ...
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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 ...
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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 ...
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1answer
27 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, ...
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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 $x\in G$
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2answers
34 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 $e$ ...
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16 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}\\ ...
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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 ...
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1answer
56 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 ...
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1answer
46 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 ...
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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 ...
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1answer
38 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 ...
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20 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 ...
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1answer
26 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 ?
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1answer
31 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 ...
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21 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 ...
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17 views

showing the inverse of an element is a generator of the non-inverse

Question: Let $G$ be a group and let $a \in G$. Prove that $\left \langle a^{-1} \right \rangle=\left \langle a \right \rangle$ Suppose $\left \langle a^{-1} \right \rangle$ so $\left ...
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2answers
32 views

find the maximum value of some Summation

Let A be a groups of some numbers: $$A = \big\{x_{1},x_{2},...,x_{n}\big\}$$ such that for any 1<= i <= n: $$0<= x_{i} <=1$$ and $$\sum_{i=0}^n x_{i} = 1$$ There is a formula for the ...
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1answer
144 views

Sharp bounds for the number of groups of order $75600$?

How can I get sharp bounds for $gnu(75600)$, the number of groups of order $75600$. I tried to determine the number of groups of order $15120$ to get a reasonable lower bound, but I quit GAP after ...
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2answers
41 views

Subgroups of $\mathbb{Z}_p^n$

Is there a nice characterization or construction to list the subgroups of $\mathbb{Z}_p^n$, that is, $\mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$ where $\mathbb{Z}_p$ is the cyclic group of prime ...
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1answer
24 views

Are there Only Two different types of group structure of order 4??

I learned in my Abstract Algebra class that there are only two forms of group structure of order 4. One is \begin{array}{c|cccc} +&0&1&2&3\\ \hline 0 & 0 & 1 & 2 & ...
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18 views

generator of a subgroup

How many subgroups does $\mathbb{Z}_{20}$ have? List a generator for each of these subgroups? By the fundamental theorem of Cyclic group: The subgroup of the the Cyclic group $\mathbb{Z}_{20}$ are ...
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1answer
13 views

Nilpotent Hall subgroup in a solvable finite group and radical condition

The following is a classic result: Let $G$ be a solvable finite group. Assume that $G$ has an abelian Hall $\omega-$subgroup $H$ and that $O_{\omega'}(G)=1$. Then $H\triangleleft G$. Is the result ...
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1answer
30 views

If $H$ is a normal subgroup of $p$-group $G$ then why is $H\cap Z(G)\neq \{e\}$? [duplicate]

If $H\neq\{e\}$ is a normal subgroup of the $p$-group $G$ Why is it that $H\cap Z(G)\neq \{e\}$? Also if the last statement is true why is it that if $|H|=p$ then $H\subseteq Z(G)$?
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1answer
34 views

Why is it that if $|G|=p^{n}$ then $|Z(G)|\neq p^{n-1}$?

I am reading a proof about why If $|G|=p^{n}$ (where $p$ is prime) then $|Z(G)|\neq p^{n-1}$? That proof says that if $|Z(G)| = p^{n-1}$ then $G/Z(G)$ is cyclic which makes $G$ abelian. My question ...
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1answer
22 views

Proof for generator of the group of integer under addition modulo

Theorem: An integer $k$ in $\mathbb{Z}_{n}$ is a generator of $\mathbb{Z}_{n}$ If and Only if $gcd\left ( n,k \right )=1$ My problem lies with proving the "If" condition and here is my ...
3
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1answer
42 views

Using Irreducible Group Characters to Count nth Roots of Group Elements

Given $n\in\mathbb{N}$, define $\tau_n(g)=|\lbrace h\in G: h^n=g\rbrace|$. Let $\chi_i,1\leq i\leq r$ be the distinct complex irreducible characters of a finite group $G$, and let ...
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32 views

Can the number of non-solvable groups of a given order be easily determined?

It can be extremely difficult to find the number of groups of a given order. But if we only want to find the number of non-solvable groups of a given order, is there an easy algorithm doing the job ...
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1answer
25 views

Is every non-solvable group a product of a set and a subgroup?

Every solvable group is a Zappa-Szep-product. Non-solvable groups need not be a Zappa-Szep-product, $A_6$ being the smallest counter-example. However, if we define ...
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25 views

Proof for generators of cyclic group [duplicate]

Theorem: Let $G=\left \langle a \right \rangle $ be a cyclic group of order n. Then $G = \left \langle a^{k} \right \rangle$ if and only if $gcd\left ( n,k \right )=1$ I've proven the "only ...
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1answer
54 views

Order of element in group theory

I have started taking a course on group theory and I have some confusion on few things If $G$ is a group, and $H$ is a normal subgroup, and $P$ is a prime number then Suppose $y\,H \in G/H$ is the ...
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1answer
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Question about notation in group theory

If you click on the link below you will find a theorem from Daniel Gorenstein's book "Finite Groups". I am not sure what is the meaning of the i'(x). What does the ' mean? http://prnt.sc/as5413 ...
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Given $G$ finite and $L(G)=G$ where $L(G)=\{g\in G\;|\;\alpha(g)=g, \forall \alpha\in \textrm{Aut}(G)\}$ . Prove that $G=1$ or $G=\mathbb{Z}_2$

Given $G$ finite and $L(G)=G$ where $L(G)=\{g\in G\;|\;\alpha(g)=g, \forall \alpha\in \textrm{Aut}(G)\}$ Prove that if $L(G)=G$, then $G=1$ or $G=\mathbb{Z}_2$. Here I already proved that $L(G)\leq ...
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29 views

Trouble understanding Sylow's Third Theorem

The statement of Sylow's third theorem in my text goes like this, Let p be a prime and let G be a group of order $p^km$, where $p$ does not divide $m$. Then the number $n$ of Sylow $p$-subgroups of ...
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42 views

For which finite groups $G$ does a finite group $H$ exist, such that $Aut(H)$ is isomorphic to $G\ $?

Given a finite group $G$, how can I check whether a finite group $H$ exist, such that $\operatorname{Aut}(H)$ is isomorphic to $G$ ? Here ...
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1answer
39 views

Distributive subgroups lattice

Let $G = \langle a \rangle \times \langle b \rangle $ ($a,b \in G$), where $ |\langle a \rangle| = n, |\langle b \rangle| = m$ and $gcd(n,m) = d > 1$. I need to show that subgroup lattice of $G$ is ...
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1answer
153 views

Number of $p$-groups with order $p^k$ and $|Z|=p$

What is known about the number of groups of order $p^k$ with $|Z|=p$ , which I denote $N(p^k)$ ? For $k=1$ , it is clear, that we have $N(p^k)=1$ : We have one group of order $p$ and since it is ...
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1answer
30 views

Center of a semidirect product

Here http://planetmath.org/node/87994 a formula for the center of the semidirect product of two groups for a given homomorphism is given. I also wonder whether the formula is correct or not. The ...
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1answer
59 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
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11 views

Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure: $$ W^TW=(P_{1}t,\cdots,P_{k}t) $$ with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...
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0answers
25 views

Sylow p-subgroups of a finite group G which are contained in some other subgroup H are also Sylow p-subgroups in H

I have found a proof to the following question but I don't quite understand it. Let $P \leq H \leq G$ and $|G|=p^\alpha m$, where $p$ doesn't divide $m$. I need to show that if $P \in Syl_p(G)$, ...