0
votes
2answers
198 views

Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
0
votes
1answer
34 views

Isomorphism of Groups (C.S.I.R)

Suppose $(F , +, .)$ is the finite Field with 9 elements. Let $G = (F , +)$ and H = (F \ {0}, .) denotes the underlying additive and multiplicative groups respectively, Then $ G \cong \mathbb Z_3 ...
5
votes
3answers
50 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
2
votes
1answer
40 views

Subgroups of $(\mathbb Z_n,+)$

The problem is to define all subgroups of $(\mathbb Z_n,+), n \in \mathbb N$. My guess is if n is prime number, then there is only trivial subgroups. If n is not prime, then I can factorize it, and ...
2
votes
1answer
89 views

How many normal subgroups does a non-abelian group $G$ of order $ 21$

How many normal subgroups does a non-abelian group $G$ of order $21$ have other than the identity subgroup $\{e\}$ and $G$? 1) 0 2) 1 3) 3 4) 7 I think option 1 is incorrect because every group ...
4
votes
1answer
135 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
0
votes
1answer
62 views

Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
1
vote
1answer
55 views

Orbit-Stabilizer Theorem proofs?

I posted 3 full questions to give context. But my main problem is the second part of the questions and how would they be answered/proved.
0
votes
1answer
55 views

Proving that Aut($S_3$) is isomorphic to $S_3$

I'm doing an exercise were I had to first prove that all automorphisms of $S_3$ induce a permutation in $X= \{ \alpha \in S_3 \, / \, $order$(\alpha) = 2\}$, which was easy enough. Now I have to ...
0
votes
1answer
30 views

Symmetry Group Regular Tetrahedron

Looking for some help of how to do this, which could also be expanded to other shapes. Thanks.
0
votes
1answer
49 views

Group Theory, Permutations.

Let $x := (175)(2436)$ and $y := (1234567)$ Compute the elements $x^{-1}yx$ and $y^{-1}xy$? And if $z := (1326745)$ for which $u$ is there an element $v$ such that $v^{-1}zv=u$. If $u$ exists find ...
1
vote
2answers
22 views

if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
1
vote
1answer
55 views

How to count generators for a cyclic group

Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
1
vote
4answers
65 views

If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

Hint: Use Fermat's Theorem that $a^{p-1}\equiv 1\mod p$ if $p \nmid a$. (I have no idea, but something in group theory should help)
0
votes
2answers
65 views

solutions of $x^2\equiv 1 \pmod p $ [duplicate]

If p is a prime, show that the only solutions of $x^2\equiv 1 \pmod p $ are $x=1$ and $x\equiv -1 \pmod p$. (from herstein's abstract algebra chapter2 section4 lagrange's theorem problem 15, this ...
1
vote
2answers
35 views

A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
2
votes
2answers
91 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
0
votes
2answers
47 views

Groups - Inversions

Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the ...
0
votes
1answer
33 views

Normal Sylow $p$-subgroup of a normal subgroup

Any hints for the following question - I am sure that I am missing something very simple here. $K$ is a normal subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $K$. If $P$ is a normal subgroup of ...
1
vote
4answers
66 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
1
vote
2answers
41 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
1
vote
1answer
32 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
3
votes
3answers
117 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
1
vote
1answer
48 views

Building a partial injective relation

Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$. Example: $A ...
1
vote
1answer
47 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
3
votes
1answer
36 views

Specify finite group satisfying two conditions

Let $G = \left\{ 0, 1, 2, 3, 4, 5, 6, 7\right\}$ be a group with the operation $\circ$, which satisfies the following conditions: \begin{align} a \circ b \leq a + b \quad & \forall a,b \in G & ...
0
votes
2answers
56 views

Find all the groups $G$ such that $|G|\leq 6$

Problem statement: Find all the groups of order at most 6. Attempt at a solution: What I thought was, if $|G|=1$, then the only possible element of the group is the neutral element. Now note that ...
3
votes
1answer
93 views

a question about fixed-point-free automorphism

Let G be a finite group with a fixed-point-free automorphism a of order 3. Prove that [x,y,y]=1 for all x,y in G.
1
vote
1answer
45 views

Composition Series of $A_4 \times S_5$

Please help me with the following question: Find the composition series of $A_4 \times S_5$ and prove that this series is indeed a composition series. Afterwards, find a group with the same ...
4
votes
2answers
210 views

Subgroups of the Klein-4 Group

Can anyone explain to me the subgroups of the Klein-4 group? I'm trying to view it this way: I want some groups that are not empty and $ab^{-1} \in H$, where $H$ denotes the subgroups I am looking ...
0
votes
1answer
202 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order 2 or that $G$ has more than one element of order 2. ...
1
vote
1answer
40 views

Identify a semidirect product $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$

I'm studying for the first time semidirect product and I'm trying to learn how to identify some of them. For example $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$ I red that, for ...
2
votes
1answer
130 views

Sylow p-subgroups, normal subgroups and the center subgroup

Let: $G$ be a finite group. $p$ be a prime number. $P$ be a Sylow-p subgroup of $G$. If $p\mid o(G)$ and for every $(a,b)\in G$, $(ab)^p=a^pb^p$, please help me prove the following: (1) ...
0
votes
1answer
74 views

Proving that there exists an element of order $p^2$ in a finite abelian group

I've been stuck on this problem for a while now. Let $G$ be finite and abelian. Suppose $\exists x \in G$ such that $x$ has non-square-free order, i.e., $|x| = p^km$ with $p$ prime and ...
2
votes
1answer
46 views

Abstract Algebra: Index of Subgroups

Here's the problem I'm working on: Prove: Suppose $H$ has index $p$ and $K$ has index $q$, where $p$ and $q$ are distinct primes. Then the index of $H \cap K$ is a multiple of $pq$. (Plus: do you ...
0
votes
2answers
81 views

Basic Abstract Algebra - Homomorphism [duplicate]

Given a homomorphism $f:G \rightarrow H$, $G$ finitely generated, what can you say about the order of $g_i$ and $f(g_i)$? I've thought about this question for a while but haven't come to a ...
1
vote
2answers
93 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
4
votes
2answers
114 views

Every element of a finite abelian group with square free order equivalence

I'm currently having some trouble with this problem: Given $G$ a finite abelian group, prove the following are equivalent: $1.$ Given any subgroup $H$, there exists a subgroup $K$ ...
1
vote
1answer
42 views

Cyclic Subgroup?

For $U(16) = \{1,3,5,7,9,11,13,15\},$ is there a simple way to find $m \in U(16)$ such that $|m| = 4$ and $|\langle m\rangle \cap \langle 3\rangle| = 2$ and $m$ is unique without listing everything ...
3
votes
1answer
165 views

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. [closed]

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. I have no idea. Give me some hints.
2
votes
1answer
143 views

Abelian subgroup in the symmetric group

Let $p$ be a prime number. Show that there is an abelian subgroup $P$ of order $p^p$ in $S_{p^2}$ such that every element in $S_{p^2}$ that isn't in $P$ does not commute with every element in $P$. ...
2
votes
1answer
95 views

Irreducible representations of group of order $pq$

There is the problem to describe dimensions of irreducible representations of a group of order $pq$, where $p$ and $q$ a distinct primes. I am doing it as follows: Suppose $p>q$. Then by the Sylow ...
1
vote
0answers
58 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
2
votes
2answers
146 views

Sylow p-subgroup of a direct product is product of Sylow p-subgroups of factors

Let $G$=$HK$ be a finite group (direct product), $P$ a Sylow $p$-subgroup of $G$. Prove that $P$ = $H'K'$, where $H'$ and $K'$ are Sylow $p$-subgroups of $H$ and $K$ respectively. I am very new ...
2
votes
1answer
117 views

Isomorphic finite abelian groups

Let $G$ and $H$ be finite abelian groups. Show that if for any natural number $n$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G$ and $H$ are isomorphic. I know, ...
1
vote
1answer
60 views

Subgroup that contains all Sylow $p$-subgroups

Let $P$ be a Sylow $p$-subgroup of a finite group $G$, for some prime $p$. Prove that if $H$ is a subgroup of $G$ that contains all Sylow $p$-subgroups of $G$, then $G = HN_G(P)$. Here's what I ...
4
votes
1answer
80 views

What's the smallest $n$ such that $D_{12}$ has an isomorphic copy in $S_n$?

I can show that $S_7$ is the smallest candidate for the property given. And, with a little calculation, I think it works out - $(1234)(567)$ and $(14)(23)(57)$ seem to generate such a subgroup. But I ...
1
vote
1answer
41 views

Finding the group generated by 2 given 3 * 3 binary matrices

Having trouble completing this exercise. I posted a few questions on subgroups generated by subsets of a group. But am still at odds on how to solve a problem of this type. The orders of the first ...
0
votes
1answer
62 views

Two questions about order of subgroups

We talking about $\mathbb{Z}_{66}\times \mathbb{Z}_{35}$. $\gcd(66,35)=1 \Rightarrow\;\mathbb{Z}_{66}\times \mathbb{Z}_{35}\;$ is cyclic. A. I need to find a subgroup with order 210, and tell how ...
2
votes
3answers
655 views

Prove that a finite abelian group is simple if and only if its order is prime.

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...