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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1answer
93 views

Is there a simple and a non-simple group with same numbers of elements of each order.

Are there finite groups $G$ and $H$ such that: $n:=|G|=|H|$. $G$ is simple. $H$ is not simple. for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ?
3
votes
2answers
179 views

Express $\alpha^{83} $ as a product of disjoint cycles

I have $\alpha$ = $(15)(37964)(8)(2)$ and am asked to express it to the power of $83$ This is what I have done so far, $\alpha ^{83} = (15)^1(37964)^3(8)(2) \: = (51)(46937) $ Am I doing it ...
9
votes
2answers
247 views

If $p$ is an odd prime, does every Sylow $p$-subgroup contain an element not in any other Sylow $p$-subgroup?

Suppose that $p$ is an odd prime. Does every Sylow $p$-subgroup of a finite group contain an element that is not contained in any other Sylow $p$-subgroup? Or does there exist a group $G$ with Sylow ...
2
votes
5answers
5k views

Expression as a product of disjoint cycles

Let $\alpha = (9312)(496)(37215) \in S_n, n \ge 9$. Express $\alpha$ as a product of disjoint cycles. I know this is probably a really easy question, but my professor didn't elaborate on how to ...
3
votes
0answers
62 views

Let G be finite and let $p$ be the smallest prime dividing $|G|$. Let $H \le G$ be of index $p$. Prove that $H$ is a normal subgroup of $G$. [duplicate]

This is a problem from Herstein that I have been stuck upon for ages. I am becoming increasingly disappointed and disillusioned about my abilities due to this problem. Let G be finite and let $p$ ...
5
votes
3answers
162 views

$G$ is a finite group and $G_1$,$G_2$ are subgroup that $G_1 \cap G_2=\{1\}$, then $|G_1|.|G_2| \big| |G|$?

$G$ is a finite group and $G_1$,$G_2$ are subgroup that $G_1 \cap G_2=\{1\}$, then $|G_1|.|G_2| \big| |G|$? If $x\in G_1$, $y \in G_2$ implies $xy=yx$ or one is normal, the statement is obvious. But ...
14
votes
1answer
377 views

The “architecture” of a finite group

I think that the aim of the finite group theory is the following: Given a generic finite group $G$, study completely the subgroup structure of $G$. There are at least two ways to achieve this ...
3
votes
1answer
95 views

Groups with transitive automorphisms

Let $G$ be a finite group such that for each $a,b \in G \setminus \{e\}$ there is an automorphism $\phi:G \rightarrow G$ with $\phi(a)=b$. Prove that $G$ is isomorphic to $\Bbb Z_p^n$ for some prime ...
2
votes
1answer
104 views

When does there exist a $g\in G$ such that $H = gH'g^{-1}$?

Suppose $H$ and $H'$ are two finite subgroups of $G$. Do there exist theorems that give conditions so that $H$ and $H'$ are conjugate, that is, there exists $g\in G$ such that $H = gH'g^{-1}$? Edit: ...
6
votes
1answer
349 views

How many finite groups (up to isomorphism) are abelian?

Do the set of finite non-abelian groups and the set of finite abelian groups have the same cardinality? In that case, is it possible to define some sort of density of finite groups and calculate the ...
2
votes
2answers
130 views

Given $v, w$ find a matrix $P$ such that $v = Pw$

How can I show that given two non-zero vectors $v, w \in \mathbb{F}_q^2$ there exists a matrix $P \in SL_2(\mathbb{F}_q)$ such that $v = Pw$?
5
votes
3answers
488 views

Abelian $p$-group with unique subgroup of index $p$

Let $G$ be a finite abelian $p$-group with a unique subgroup $H$ of index $p$. It is a fact that $G$ is cyclic. This can be deduced from the classification theorem for finite abelian groups by writing ...
4
votes
4answers
144 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
5
votes
3answers
402 views

subgroup of order $11$ lies inside $Z(G).$

I need help to solve this problem: Let $G$ be group of order $231.$ we need to show that the subgroup of order $11$ lies inside $Z(G).$
1
vote
2answers
122 views

Finding the number of orbits

How many orbits are there of $(12)(25)$ in $S_{5}$? Considering the permutation $(12)$, it has $4$ orbits and is as follows: $\{\{1,2\},\{3\},\{4\},\{5\}\}$ and (25) also has 4 orbits and is also ...
5
votes
1answer
105 views

For which triples $(i,j,k)$ is the presented group finite?

I was wondering when is the group $$\langle a,b \mid a^i, b^j, (ab)^k \rangle$$ finite? Here are some examples: Tetrahedral, Octahedral and Icosahedral groups: $\langle s,t \mid s^2, t^3, (st)^3 ...
4
votes
2answers
85 views

A problem of finite group (related to Sylow's theorem?).

First of all, let me describe the problem that I try to solve: Let $p$ be a prime, $G$ a finite group, $P$ a $p$-Sylow subgroup of $G$, and $L$ a set of all elements of $G$ which its order is ...
5
votes
1answer
126 views

What is the centralizer of the Young symmetrizer?

I have read a lot about idempotents, several important facts were about central idempotents. Now, the Young symmetrizer is a constant away from an idempotent, but I don't think it's central. ...
1
vote
3answers
124 views

About minimal subgroup of a finite group.

Let $G$ be a finite group such that $G=PQ$ where $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is a normal Sylow $q$-subgroup of $G$. I need to prove the following: (1) If $H/Q$ is a subgroup of $PQ/Q$ ...
6
votes
1answer
118 views

Do orders uniquely describe a group? [duplicate]

Possible Duplicate: Three finite groups with the same numbers of elements of each order With any finite group $G$ I can associate a multiset $S_G = \{\text{ord}(g) : g \in G\}$. Is the map ...
0
votes
0answers
54 views

A question on a group of order 28 [duplicate]

Possible Duplicate: A group of order 28 with a normal Sylow 2-subgroup is abelian Let $G$ be a non abelian group of order 28. Is it true that $G$ contains a normal subgroup of order 4?
3
votes
1answer
118 views

Constructing Steiner System $S(5,8,24)$ and Mathieu Group $M_{24}$

Let $a_0,...,a_{2^{24}-1}\in \{0,1\}^{24}$ be a sequence such that $a_k$ is the binary representation of $k$. i.e. $$a_0=(0,...,0,0,0,0)$$ $$a_1=(0,...,0,0,0,1)$$ $$a_2=(0,...,0,0,1,0)$$ ...
6
votes
2answers
126 views

$N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.

The following is a problem from the Berkeley Problems book. Let $G$ be a group with three normal subgroups $N_1$ , $N_2$, and $N_3$. Suppose $N_i \cap N_j = \{ e\}$ and $N_iN_j = G$ for all ...
2
votes
1answer
66 views

Existence of a “$pq$-free” subgroup of a solvable group.

Let $G$ be a finite solvable group and denote by $\pi(G)$ the set of prime divisors of $|G|$. Suppose there is a $p\in \pi(G)$ for which there is an element of order $pq$ in $G$ for every $q\in ...
2
votes
2answers
142 views

Constructing $M_{24}$ using Steiner Systems.

I'm trying to (re)construct the Mathieu Group $M_{24}$ using Steiner systems. I'm not so familiar with t-designs and Steiner systems. I have just the definition of t-designs, Steiner systems and the ...
8
votes
2answers
313 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
4
votes
1answer
117 views

Exponent of the alternating group.

The exponent of a group $G$, denoted $\text{Exp}(G)$, is the smallest $n\in \mathbb{N}$ such that $x^n=1$ for every $x\in G$. This page says that ...
2
votes
0answers
75 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
0
votes
1answer
45 views

Prime divisors of irreducible degrees

I would like to ask that : assume that $N$ is a normal subgroup of a finite group $G$, if $p$ is a prime divisor of $\chi(1)$ for some $\chi\in Irr(N)$, does it imply that $p$ divides $\varphi(1)$ for ...
0
votes
1answer
98 views

Semi directProduct and Maximal subgroup in gap [closed]

Let $P$ be a quaternion of order 8 and $Q$ a cyclic group of order 9 and $G=[p]Q$, a semidirect product ($P$ is normal in $G$). Let $M$ be a maximal subgroup of $G$ such that $Q<M$. I want to ...
2
votes
2answers
739 views

Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .

I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p). I should note that by simple I mean ...
2
votes
1answer
82 views

Burnside's Action Equivalence Criterion

Theorem: Actions of a finite group G on finite sets $X$ and $Y$ are equivalent if and only if $|X^H|=|Y^H|$ for each subgroup $H$ of $G$ I've proved the finite case inductively but I'm wondering ...
4
votes
1answer
100 views

Is there a simple construction of a finite solvable group with a given derived length?

Given some integer $n$, is there an easy way to construct a finite solvable group of derived length $n$? It would seem that given a solvable group of length $n-1$, one should be able to form the ...
4
votes
0answers
97 views

References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
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0answers
219 views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
2
votes
1answer
65 views

Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .

let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,...,n^2\}$ with distinct entries. let $T$ be the set of all permutations of $\mathbf{A}$ which swap the entry 1 with ...
8
votes
1answer
298 views

Generic properties of $p$-groups

I have the impression that most people in algebra believe the following statement to be true, but I have no reference for it. Fix a natural number $n$. Consider for each prime $p$ the set of all ...
10
votes
5answers
750 views

Is $\mathbb Z _p^*=\{ 1, 2, 3, … , p-1 \}$ a cyclic group?

In undergraduate course, the two groups which are most frequently used may be $$\{ 0, 1, 2, ... , p-1\}$$ and $$\{ 1, 2, ... , p-1\}$$ where $p$ is a prime. The first one is a group under addition ...
1
vote
3answers
2k views

Calculate the order in cycle notation

Please help calculate the order of x and y. Let x and y denote permutations of $N(7)$ - Natural numbers mod 7 . Cycle notation: $$x= (15)(27436) $$ $$y= (1372)(46)(5)$$ Thanks
2
votes
0answers
113 views

How many rings have four elements? [duplicate]

Possible Duplicate: There are at least three mutually non-isomorphic rings with $4$ elements? How can I prove how many rings are commutative, unitary with 4 elements? (obviously ...
1
vote
2answers
285 views

General approach to determining if a subset is a subgroup if it has finite order

I am a little confused as how to approach problems that ask whether a subset is a subgroup given that it has the property of being of finite order e.g. in the case for $GL(N,\mathbb{R}$). What ...
14
votes
1answer
364 views

What are all conditions on a finite sequence $x_1,x_2,…,x_m$ such that it is the sequence of orders of elements of a group?

My Definition: The finite sequence $x_1,x_2,...,x_m$ of nonnegative integers, is said to be generated by the finite group $G$ iff $n:=|G|=x_1+x_2+...+x_m$. $n$ has $m$ divisors. if ...
2
votes
0answers
389 views

How to find the number of orbits

In our lecture notes, i am confused on how to determined the number of orbits with the given permutations in S4. If i consider the permutation (12) . I can also rewrite it as cycle decomposi- tion and ...
5
votes
3answers
2k views

prove : if $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $|G|$ then any subgroup of index $p$ is normal

Prove: If $G$ is a finite group of order $n$ and $p$ is the smallest prime dividing $ |G| $ then any subgroup of index $p$ is normal where $ |G| $ is the order of $G$ This is a result in ...
5
votes
1answer
118 views

Existence of a finite group having a certain kind of 2-dimensional representation.

Is there a finite group $G$, an element $c$ of order 2 in $G$, and an irreducible 2-dimensional complex representation $\rho$ of $G$ such that all the following are true: 1) $\rho(c)$ has trace zero ...
4
votes
2answers
329 views

On Symmetric Group $S_n$ and Isomorphism

I use Abstract Algebra by Dummit and Foote to study abstract algebra! At page 120, section 2 in chapter 4, there is a great result form my point of view which proves that, for any group $G$ of order ...
1
vote
1answer
45 views

According to my solution there should be more fixed points…

I have solved the following exercise: A group of order $55$ acts on a set of order $18$. Then there are at least $2$ fixed points. But according to my solution, there should be at least 3 fixed ...
8
votes
4answers
306 views

Is there an abelian and a nonabelian group with same numbers of elements of each order.

Are there finite groups $G$ and $H$ such that: $n:=|G|=|H|$. $G$ is abelian. $H$ is nonabelian. for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ? I know two finite ...
6
votes
3answers
260 views

Smallest possible subgroup of $\,D_6\,$ containing two elements of $\,D_6\,$

Let $\,G = D_6;\;$ let $\,H\,$ be the smallest subgroup containing the elements $\,r^2s\,$ and $\,sr^2.\;$ List all the elements in $\,H\,$ and explain. My intuition leads me to $H = \{ r^2s, ...
3
votes
1answer
69 views

Semidirect Product $(A_{5} \times A_{5}) \rtimes Z_{2}$

Consider the group $G=(A_{5} \times A_{5}) \rtimes Z_{2}$, where $A_{5} \times A_{5}$ is the normal subgroup of $G$. $Z_{2}$ acts by swapping the two copies of $A_{5}$. I checked with gap that $A_{5} ...