# 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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### Normal subgroups of $A_5$ must contain a 3-cycle.

I am trying to prove the simplicity of $A_5$ by showing that every non-trivial normal subgroup $H$ contains a 3-cycle, and therefore is all of $A_5$ since the 3-cycles all belong to one conjugacy ...
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### Non-abelian Order of $6$ is isomorphic to $S_3$ [on hold]

I know that it's duplicate, but , How can I prove it? I know that must be element "$a$" of order 2, and element "$b$" of order 3. What is the next step? In fact, from what I search it is claimed that ...
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### Can someone find example for it? [on hold]

Example for group that for any element the order is $2$ apart of $e$. (without $Z_2+Z_2+Z_2+\cdots+Z_2$)
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### It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold]

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1
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### Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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### $G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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### Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
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### Linear representations of projective groups

Does the projective linear group $PSL_2(\mathbb{R})$ admit faithful linear representations? In other words, does there there exist a homomorphism $SL_2(\mathbb{R}) \to GL_n(\mathbb{R}),$ for some $n$, ...
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### Group Theory: Orders of Elements

Have been given following problem. Let G be group and x $\in G$. Prove that $x^2 = e$ if and only if x is of order one or two. My response. If order of x is 1, then by definition of order of an ...
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### groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
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### Why isn't the orbit-stabilizer theorem obvious?

The title of this post paraphrases the title of a great blog post by Timothy Gowers, where he argues that those who think that the fundamental theorem of arithmetic is obvious are almost certainly ...
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### Prove all products of two disjoint 2-cycles are pairwise conjugate in $A_{n}$

Let $n \geq 4$. Prove that in the group $A_{n}$ of parity-even permutations of $\{1,2,...,n\}$, all products of two disjoint 2-cycles are pairwise conjugate. This is a past exam question for a ...
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### Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
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### Determining all the elements of S4?

What is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I tried ...
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### All 3-cycles are pairwise conjugate.

In my lectures on Group Theory our lecturer claimed the following: All 3-cycles are pairwise conjugate. He then went on to prove this but I am struggling with understanding his proof. I will try and ...
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 to ...
what is the dimension of $O(n) \setminus \mathbb{R}^n$ for $n \geq 2$? how to calculate this? $n - \frac{n(n-1)}{2}$ does not work well.