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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2answers
441 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
77 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
88 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 ...
3
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0answers
84 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
184 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
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1answer
61 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
264 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 ...
6
votes
5answers
527 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 ...
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vote
3answers
1k 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
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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
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2answers
233 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 ...
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1answer
351 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
309 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 ...
4
votes
3answers
1k 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
111 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
285 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
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1answer
42 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
270 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
241 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
68 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} ...
3
votes
1answer
163 views

On Group action and blocks of subgroups of the symmetric Group

this exercise is from Dummit and foote , page 117 , # 7.d prove : a transitive group $G$ is primitive on $A$ iff for each $a \in A$ , the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$ ...
6
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2answers
515 views

Finding all groups of order $2p^2$

How can one find all groups of order $2p^2$ up to isomorphism, where p is an odd prime. I know there's 5 groups of order 18. The p-Group $P$ is normal and abelian.
4
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1answer
72 views

Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler

I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense: Each element of $H$ can be represented by one or a few elements of ...
4
votes
4answers
365 views

Every group of order 203 with a normal subgroup of order 7 is abelian

This is a question from Dummit & Foote. Let G be a group of order 203. Prove that if H $\leq$ G is a normal subgroup of order 7 then H $\leq$ Z(G). Hence prove that that G is abelian. My ...
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vote
1answer
154 views

Is there any normal subgroup of $S_n$? [duplicate]

Possible Duplicate: Normal subgroups of $S_N$ I wonder if there is any normal proper subgroup of $S_n$? If yes, give an example.
9
votes
1answer
168 views

Alternating group $A_n$ where $n\geq 5$

I tried so much to prove the following fact about the alternating groups $A_n$, $n\geq 5$ but I couldn't prove it. Any answer or hint will be appreciate; Any maximal ...
1
vote
1answer
104 views

Infinite index subgroups of $SL(3,\mathbf{Z})$

Finite index subgroups of $SL(3,\mathbf{Z})$ are well-know (at least we know that they are congruence subgroups). But I wasn't able to find reference on infinite index subgroups. Does someone knows ...
3
votes
2answers
358 views

Inner automorphism of a subgroup

Dummit & Foote defines inner automorphism as : Let $G$ be a group and let $g \in G$. Conjugation by $g$ is called an inner automorphism of $G$. Later they say: If $H$ is a normal ...
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2answers
103 views

group of order $198$

could any one tell me how many non isomorphic groups are there of order $198$? and how many elements have order $11$ in them? A group which first come to my mind is $Z_{198}$, I really do not know, ...
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3answers
206 views

Existence of Subgroups of Index $p$

When I was attempting (and ultimately not succeeding) to answer this question I wanted something in group theory to be true, which I didn't think was. Namely if $G$ is a finite group and $p$ was a ...
1
vote
2answers
59 views

Exhibit a cycle with a power having a given cycle shape

Problem: If $$τ=(1~ 2)(3 ~4)(5~ 6)(7~ 8)(9~ 10)$$ determine whether there is an $n$-cycle $σ,~~n≥10$ such that $τ=σ^k$ for some integer $k$. I found let $σ=(1~ 3~ 5~ 7 ~9 ~2~ 4 ~6 ~8~ 10)$, then ...
2
votes
1answer
67 views

Find all the elements in Dih(2n), n odd, which commute with all other elements

Look at this page: http://crazyproject.wordpress.com/2010/01/08/find-all-the-elements-in-dih2n-n-odd-which-commute-with-all-other-elements/ What is the “$q$” in the answer? And what "$0\leq a_0$" ...
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3answers
162 views

Finite group and subgroups

$G$ is finite group. $A,B$ are subgroup of $G$. $A \nsubseteq B$. I need to prove that $|A\cap B|$ $\le$ $\frac{|A|}{2}$. I think it might have something to do with Lagrange but I can't find how it ...
0
votes
1answer
74 views

Simple faithful $KA$-modules over finite fields

Let $V$ be a vector space of dimension $n$ over finite field $K=\mathbb{F}_q$ and let $A$ be an abelian group such that $V$ is simple, faithful $KA$-module. Then $A$ is cyclic. Moreover, for every ...
1
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1answer
148 views

Isomorphism types of simple $KG$-modules with $G=S_3$

I am trying an exercises on determining all isomorphism types of simple $KG$-modules with $G=S_3$, the symmetric group of degree $3$. If $K$ is algebraically closed then we can use the following ...
3
votes
1answer
88 views

Chief series of a group

I need help in checking some reasoning in an answer. Let $G$ be a group with order 180. Suppose G is a group with chief series $G = G_0 \geq G_1 \geq G_2 \geq \cdots \geq G_r = \{1\}.$ What are the ...
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3answers
1k views

Problem from Herstein on group theory

The problem is: If $ G $ is a finite group with order not divisible by $ 3 $, and $ (ab)^{3} = a^{3} b^{3} $ for all $ a,b \in G $, then show that $ G $ is abelian. I have been trying this for a ...
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3answers
147 views

Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.

"Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$." Exercise 27.5 from "Groups and Symmetry" M.A.Armstrong. This should be an easy ...
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vote
2answers
82 views

Group of distance preserving transformations of the plane is isomorphic to $\mathbb{Z} \ltimes \mathbb{Z}$

"Let $G$ be the group of distance preserving transformations of $\mathbb{R^2}$ which is generated by $(x,y)\mapsto(x+1,y)$and $(x,y)\mapsto(-x,y+1)$. Prove that $G$ is isomorphic to the semidirect ...
2
votes
2answers
156 views

Exponent of a direct product of cyclic groups

I have an answer to a homework question that I am not sure is correct. The question is show that if $G \cong C_{n_1} \times C_{n_2} \times \cdots \times C_{n_k}$ for positive integers $n_1, n_2, ...
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1answer
520 views

Which families of groups have interesting formulas for the number of elements of given order?

Suppose that $G$ is a group and that $n$ is a positive integer diving the order of $G$. Let $f_n(G)$ be the number of elements satisfying $x^n = 1$ in $G$. According to a theorem of Frobenius, then we ...
0
votes
2answers
164 views

A question about groups and subgroups.

I am working from these lecture notes. For this example, Example. List the elements of the cyclic subgroup of $\mathbb{Z}_8\times \mathbb{Z}_{15}$ generated by $(6,10)$. $$ ...
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2answers
165 views

A problem about group of order $p(p+1)$.

Let $G$ be a group of order $p(p+1)$ where $p$ is an odd prime and $n_{p}(G) = |\text{Syl}_{p}(G)| > 1$. The problem is to count the number of elements of $G$ that do not have order $p$. The ...
3
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0answers
95 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
0
votes
2answers
92 views

Principal ideal ring if and only if it is both a Noetherian ring and a Bezout ring.

How can I prove it? Principal ideal ring if and only if it is both a Noetherian ring and a Bezout ring. I think that The fact that a Noetherian Bezout ring is a principal ideal ring follows by a ...
4
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1answer
140 views

On automorphisms group of order $p^n$

Let $G$ be a finite group, such that $\mid Aut(G)\mid=p^n$. Then prove $G$ is p-group or $G\cong P\times C_{2}$, where $P$ is a p-group. Thank you
3
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2answers
330 views

Non-Abelian simple group of order $120$

Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
5
votes
1answer
132 views

Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$)

A question in group theory: Let $ G = M \times N $ be the direct product of $ 2 $ normal subgroups. If $( | M | , | N | ) = 1 $ then Aut($G$) $\simeq$ Aut($M$) $ \times$ Aut($N$). I proved that ...
4
votes
1answer
581 views

Order of an element in the factor group divides order or element

Let $N$ be a normal subgroup of a finite group $G$, and $a \in G$ is an element of order $o(a)$. Prove that the order $m$ of $aN$ in $G/N$ is a divisor of $o(a)$. Here what I did: ...
2
votes
1answer
104 views

Direct product of group order 2

If $P$ is group of order 2, how many subgroups (trivial and proper) has the group $P \times P \times P$? Labelling the elements of $P$ to be $e$ and $a$, list the proper subgroups.