# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 & ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $K$ ...
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### 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 ...
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### 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.
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### 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 $P$. ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...