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Questions tagged [finite-groups]

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

The questions that you can ask under this tag are of the same type that the questions related to the tag : homomorphisms of finite groups, representation theory, structure of finite groups...

You may refer to this Wikipedia entry for an introduction to this topic.

11774 questions
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Prove that if |G|=132 then G cannot be simple

Okay so I have done this but I would like a heads up if it is enough to prove it. $132=2\cdot 2\cdot 3\cdot11=2^2\cdot3\cdot11$ Let us assume that G is simple. Then from Sylow's theorem, we can say $$n_2\in\{1,3,11,33\}$$ $$n_3\in…
ZZS14
  • 819
14
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2 answers

How many automorphisms of $S_n$ take transpositions into transpositions?

I need to show that an automorphism of $S_n$ which takes transpositions to transpositions is an inner automorphism. I thought it could be done by showing that such automorphisms form a subgroups $H\le Aut(S_n)$, that $Inn(S_n)\subset H$ and that…
Artem
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13
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A lot of even elements

Back when I was an undergraduate we were asked to name a finite noncommutative group such that more then half the elements have order two. Furthermore, we know that every group such that every element different from the identity has order two must…
Severin Schraven
  • 19,179
12
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4 answers

Is there an abelian and a nonabelian group with same numbers of elements of each order.

Are there finite groups $G$ and $H$ such that: $n:=|G|=|H|$. $G$ is abelian. $H$ is nonabelian. for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ? I know two finite abelian groups with the same numbers of elements of…
user59671
10
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3 answers

On simple groups $G$, where $2\mid |G|$, $4\not\mid |G|$

The (old) exam I'm looking at has the following problem: Suppose the order of $G$ is even, but is not divisible by $4$. Prove that $G$ is not simple. A group with $2$ elements is clearly a counter-example to that. Are those the only…
user57159
10
votes
2 answers

Finding all groups of order $2p^2$

How can one find all groups of order $2p^2$ up to isomorphism, where p is an odd prime. I know there's 5 groups of order 18. The p-Group $P$ is normal and abelian.
user59671
10
votes
1 answer

in a group of size n, any n-th power of a subset is a subgroup?

Is this true? If $G$ is a group of size $n$, and $X$ is a non-empty subset of $G$ then $X^n$ is a subgroup of $G$? By $X^n$ I mean the set of all products of length $n$ from $X$.
reza
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8
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Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.

Suppose that I have a group $G$ of order $p^{2}q$ for two distinct primes $p$ and $q$. I need to first show that one of its Sylow subgroups is normal. I start by letting $H$ be a Sylow $p$-subgroup and $K$ be a Sylow $q$-subgroup. If $K$ is not…
roo
  • 5,598
8
votes
1 answer

Generators for $S_n$

This problem is from Dixon's book telling that; if $x$ is any nontrivial element of the symmetric group $S_n$ and $n≠4$, then there exists an element $y\in S_n $ such that $S_n=\langle x,y\rangle$. Honestly,the reference which Dixon referred to, is…
Mikasa
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8
votes
1 answer

"very noncommutative" groups

From group axioms, any $x$ in a group commutes with the identity, with itself, and with its inverse. By a "highly noncommutative" group is meant one for which these are the only cases of commuting elements. I found that $S_3$ (permutations of…
coffeemath
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7
votes
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How to test whether two group presentations are isomorphic

Suppose I have two presentations for groups: $\langle x,y|x^{7} = y^{3} = 1, yx = x^2y\rangle$ and $\langle x,y|x^{7} = y^{3} = 1, yx = x^4y\rangle$ What is the standard approach to deciding whether the presentations are isomorphic? I'm working…
roo
  • 5,598
6
votes
1 answer

Condition for conjugate subgroups

Let $G$ be a finite group and denote by $Z$ its centre. Let $H,K \leq G$ be two conjugate subgroups, i.e. there exists some $g \in G$ such that $K^g=H$. First of all, we get as an observation that $$K \cap Z=H^g \cap Z=H^g \cap Z^g=(H \cap Z)^g=H…
the_fox
  • 5,805
6
votes
3 answers

Centre of non-Abelian Group G, Order 6, is Trivial

Claim: Let $G$ be a non-Abelian group, $\lvert{G}\rvert=6$. Then $Z(G)=1$. Attempt so far: We know by Lagrange's Theorem that, since $Z(G)\leq G$, we have that $n:= \lvert G\rvert\in\{1,2,3,6\}$, and that since the centre is Abelian that $Z(G)\neq…
Ben
  • 460
6
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1 answer

How many homomorphisms are there from $\mathbb{Z}_{n}$ into $S_{n}$?

I would like to know how many homomorphisms there are from $\mathbb{Z}_{n}$ into $S_{n}$? If $n=2$ or $n$ is odd, I think that there are $(n-1)!+1$. I am counting those cycles of order $n$, when $n$ is odd and adding to them the trivial…
user23505
5
votes
2 answers

Any finite group is a subgroup of an orthogonal group

Prove that any finite group of order $n$ is isomorphic to a subgroup of $\mathbb{O}(n)$, the group of $n\times n$ orthogonal real matrices. Attempt: Let $G$ be a group of order $n$. Then $G$ is isomorphic to a subgroup of the symmetric group…
Eccau
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