# Tagged Questions

35 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### Generated group of 2 element of order 2

Let $G$ be a finite group that is generated by $\alpha,\beta\in{G}$ of order 2, such that their product isn't of order 2. Show that $G$ is isomorphic to $D_n$ for some n.
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### Existence of a subgroup with order 3 in a group with order 6

Let $G$ be a group of order 6. Why does $G$ has a subgroup of order 3 even if $G$ isn't cyclic? I've tried using to use negation and assume all elements in $G$ have an order of 2 or 1 but I ...
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### Find the number of permutations in $S_n$ containing fixed elements in one cycle

Find the number of permutations in $S_n$ such as 1 and 2 belong to one cycle.
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### No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
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### If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.

Let $H$ be a subgroup of a group $G$, and let $\rho :H\to GL(V)$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is at least ...
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### If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$. [duplicate]

Let $H$ be a subgroup of a group $G$, and let $\rho:H\rightarrow GL(V )$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is ...
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### Constructing elements that meet order requirements in finite group

Let $n$ be an integer. Construct a group $G$ containing two elements $a$ and $b$ of order 2 whose product is of order $n$. My attempt at a solution: If we have that $n$ is a number such that $n-1$ ...
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### If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.

I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it ...
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### Let $G$ be an abelian group such that $|G|$ is an odd integer. Show that the product of all the elements in $G$ is $e.$ [closed]

How do you show the product of the elements in $G$ is $e$?
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### $G/N$ has an element of order $m$ so, $G$ also has an element of order $m$

I swear to God, I've done about a trillion of these sorts of problems and I still don't know how to prove them. Is it as easy as saying that $G/N$ is a subgroup of $G$, and therefore $G$ contains this ...
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### If $G$ is a finite group then … [duplicate]

Let $G$ be a finite group. Let $I(G)$ be the set of elements of $G$ that have order 2. Suppose $|I(G)| \ge \frac 3 4 |G|$. Let $x \in I(G)$. We note $I_x(G)$ the subset $\{xg \in I(G)\}_{g \in I(G)}$. ...
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### Show every automorphism of a specific group arises the same way.

I have a group $G_n = U(n) \times \mathbb{Z}_n$ with the operation $(a,x)(b,y) = (ab, ay+x)$ where $U(n)$ is the multiplicative group of integers modulo n and $\mathbb{Z}_n$ is the additive group of ...
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### Number of groups of order $p^n$, where $p$ is prime

for $n=1$, it is cyclic. so, the number is $1$ for $n=2$, it is Abelian. so, the number is $2$ for $n\geq 3$, I don't know. Can you recommend a book or link which can be helpful for understanding ...
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### The group of rigid motions of an icosahedron.

Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$. Can you help me to prove this? What I have done is shown that the order of the group of rigid motions of icosahedron is ...
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### Prove that the order of any element in the additive group of integers modulo n is a divisor of n.

I am not opposed to struggle but I have been on and off of this problem for three days and need to present the proof tomorrow. I am thinking that because I know for any element in the additive group ...
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### Abstract Algebra group question

If $G$ be a finite group of $l$ elements. Suppose that $a$ belongs to $G$, and $\mathrm{ord}(a)=k$,can $k>l$? I think $k$ can't be bigger than $l$, because $k$ should equal $l$.
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### Show that G must have a subgroup of order p. [duplicate]

Let $G$ be a group of order $p^2$, where $p$ is a prime. Show that $G$ must have a subgroup of order $p$. Not really sure how to show this.
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### Counting the Elements of a Finite Group [closed]

Let $G = \mathbb Z_6 \times \mathbb Z_4$, and find $[G:H]$ for: a) $H = \{0\} \times \mathbb Z_4$ b) $H = \langle 2\rangle \times \langle 2\rangle$
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### Group of order 24 with no element of order 6 is isomorphic to $S_4$

Proposition: Given a group $G$ with $|G|=24$ such that $\nexists g\in G$ with $|g|=6$, then $G\cong S_4$. I understand methods you can employ to deduce the number of Sylow $p$-groups in $G$ by ...
I am working on a homework problem (so don't just give me the answer) from Herstein's Topics in Algebra, which goes as follows: If $G$ is a finite group, show that the number of elements in the ...