Tagged Questions

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A group of order $120$ has a subgroup of index $3$ or $5$ (or both)

What I have tried that number of $2$-sylow subgroup can be $1,3,5$ or $15$.I have solved the problem when the number of $2$-sylow subgroup is $1,3,5$. But I am not able to solve it for $15$. Any help ...
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Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
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Finding commutator of $\mathbb D_n$ and $\mathbb S_4$

I have to find $[G,G]$ (the commutator of $G$) for the the dihedral group, $\mathbb D_n$ ; and the symmetric group, $\mathbb S_4$. What tools, theorems could I use in order to find these two ...
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left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
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Cardinality of $\text{Aut}(G\times G)$

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
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If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
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Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
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The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
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Semidirect Product variations

Is there two semidirect products of order $16$ with $C_2$ normal subgroup? How many groups of the form $C_4 \times C_2$ : $C_2$ are there ? Is it one expression for two groups ? or more?
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The only group of order $255$ is $\mathbb Z_{255}$ ( Using Sylow and the $N/C$ Theorem)

Let $G$ be a group such that $G =255 = 3.5.17$. Let $H$ be a sylow $17$ sub group of $G$. Then, by Sylow's theorem : the number of sylow $17$ sub groups can be $n=1,18,35,...$ . Since, $n$ should ...
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Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$ where $cl ~(a)$ refers to the conjugacy class of $a$. The proof in the book which I am reading asks to prove ...
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A question on cyclic group with finite order

I have trouble proving the following statement: Suppose that $H$ is a finite group with order $n$ and $e$ is the identity element of $H$. For an arbitrary positive integer $d$ satisfing ...
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$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group

Let $G$ be a finite abelian group. Prove that $(a_1 a_2 \cdots a_n)^2=e$. My proof: $$\forall a \in \{a_1,a_2,\cdots,a_n\} \exists ! a^{-1} \in \{a_1,a_2,\cdots,a_n\}:a^{-1}a=aa^{-1}=e$$ hence, ...
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Group of even order must contain $a:a=a^{-1}$ $(a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
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How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $\ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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How do we know the classification of finite simple groups is finished?

I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex ...
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Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
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Motivation and intuition of double cosets

In Dummit and Foote, there was a problem on double cosets. Let $H$ and $K$ be subgroups of $G$ and $x\in G$, $HxK$ is the double coset of $x$. I can prove the double cosets partitions $G$. However, I ...
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Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
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Normalizer and centralizer of abelian subgroups of a group are equal [closed]

I have a question: Let $G$ be a finite group. If for each abelian subgroup $H$ of $G$ the centralizer and the normalizer of $H$ are equal, that is, $C_G(H)=N_G(H)$, prove that $G$ is abelian ...
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Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
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Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
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Minimal normal subgroup in a finite group

Let $G$ be a finite group and $G=HK$ such that $H<G$. How can we show that if $K$ is an abelian minimal normal subgroup of $G$, then $H$ is a maximal subgroup of $G$ and $H\cap K=1$?
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Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
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Cardinality of Centralizer of some element

Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?
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Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where ...
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A finite divisible group is trivial

I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?
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Show that there is just one subgroup $H \subset S_4$ such that $[S_4:H] = 2$

I had to show that $S_4$ has one subgroup of index $2$. Below, you'll find what I tried to do so Of course, $A_4$ is a subgroup index $2$. To show that there is not another subgroup with the same ...
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Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
Let $G$ be a group that has a subgroup of size $d$ for every $d$ that divides $|G|$. Must $G$ be abelian? It can be shown using complete induction that the converse of the above statement is true, ...