# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
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### Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
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### Number of conjugacy classes in a finite non-abelian simple group

Can we say that every finite non-abelian simple group has at least 4 non-identity conjugacy classes?
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Prove that if H has finite index n then there is a normal subgroup N of G with $N \subset H$ and [G:N]$\le$n!. I tried to solve the problem but could not done exactly. Since [G:H]=n , Let A={$a_i ... 1answer 84 views ### How do I construct the multiplication of a quotient group? The question is: If$G$is the group of all nonzero real numbers under multiplication and$N$is the subgroup of all positive real numbers, write out$G/N$by exhibiting the cosets of$N$in$G$, ... 1answer 168 views ### There are no simple groups of order$27p$Statement Show that there are no simple groups of order$27p$for any$p$prime number. I got stuck with this problem, I'll write what I've done so far: Suppose$G$is a group with$|G|=3^3p$,$p$... 0answers 61 views ### Irreducible permutations in$S_n$Let$n$be an integer$\geq 2$, let$\tau \in S_n$and let$X$be a nonempty subset of$\{1,2,\ldots,n\}$. Say that$\tau$fixes$X$if$\tau(X)=X$, and say that$\tau$acts irreducibly provided the ... 1answer 121 views ### Nonabelian group of six elements What is an example of a six-element group that is not abelian? I can't think of any. It is very possible that I am overthinking this. Thank you for any help. 1answer 378 views ### Why is a monoid with right identity and left inverse not necessarily a group? [duplicate] This problem is from Herstein's 'Topics in Algebra'. I've thought about it a bit but haven't come up with much. Let$G$be a non-empty set with an associative product which also satisfies:$\exists ...
Let $G$ be a group and $a$ and $b$ be in $G$ with $a$ of order 11 and $b$ of order 3. Show that the order of $G$ is at least 33. I'm trying to do this from first principles. Obviously with lagrange, ...
### Proving a defined group $(G,*)$ is isomorphic to $(\mathbb{R},+)$
I am studying abstract algebra and I have this question: Let $G=${$a\in\mathbb{R}|-1<a<1$} Defined an operation $*$ in $G$ with $a*b=\frac{a+b}{1+ab}$ for all $a,b \in G$ Show that $(G,*)$ ...