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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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Order of elements of the automorphism group is less than the order of the group.

Let $σ ∈ Aut(G)$, where $G$ is a nonotrivial finite group. Then show that the order $o(σ)$ of $σ$ is less than $|G|$.
DK26
  • 399
4
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If $G$ has only one subgroup of order $n$, then that Subgroup is Normal

How can I show that if some group $G$ has only one subgroup $K$ of order $n$, then $K$ is a normal subgroup? Would that mean that it only has one subgroup total? If so then I guess that makes sense.
user82004
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The structure of a finite group of order 40320

I would like to know the structure of the group $G$ where $G$ is a non-split extension of a cyclic group of order 2 by the simple group $PSL(3,4)$. (the cyclic group is normal). Can anyone help me to introduce this group to GAP?
Tina
  • 281
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subgroup of group of order $p^2$

Let $p\ge5 $ be a prime. Then $\mathbb{F_p\times\mathbb{F_p}}$ has atleast five subgroups of order p. Every subgroup of $\mathbb{F_p\times\mathbb{F_p}}$ is of the form ${H_1\times{H_2}}$ where ${H_1 , {H_2}}$ are subgroups of…
dipali mali
  • 555
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Is G a CLT group if its order is product of distinct prime?

Let G be group whose order is square-free. Is G always CLT group? Trying to apply Hall's theorem but not conclusive.
SSSM
  • 117
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What information can one obtain from the size of a finite group?

Given $n = \lvert G \rvert$ then the factorization of $n$ can sometimes give information about the group $G$. For example if $n$ is prime we know that the group is the cyclic group $C_n$. If $n$ is the power of a prime then we know that $G$ is…
Marc Bogaerts
  • 6,197
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For every $g\in G$ there exists an $h\in G$ such that $g = h^3$

Let G be a finite group whose order is not divisible by $3$. Show that for every $g\in G$ there exists an $h\in G$ such that $g = h^3$.
ram
  • 909
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The order of a group

Let G be the group defined by the following generators and relations: $G = \left\langle a, b, c \mid ab = ba, ac = ca, bc = cba, a^{3} = 1, b^{3} = 1, c^{3} = 1\right\rangle.$ Show that $\langle a\rangle$ is normal in $G$ and $|G| = 27.$ I know how…
finiteness
  • 737
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How can I identify a group given its multiplication table?

Given the group generated by the matrices $$\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix},~\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$ I get a group of order 24, as I have calculated, but which of the…
ArieBos
  • 51
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Groups of Order $210$

By the "$2n$-test", proving that a group of order $210$ cannot be simple. Is there another way to prove this? Would you use Sylow Theory?
user137957
  • 139
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In a non abelian group of order $p{^4}$ Quotient of center by commutator is abelian

Let $G$ be a non abelian group of order $p{^4}$,$p$ is a prime.Let $N$ be a normal subgroup of $G$ with |$N$|=$p$ and $G/N$ is abelian.Then prove that $N$ is a subgroup of $Z(G)$ and $Z(G)$/$N$ is a cyclic group. So far what I get as…
Panja
  • 107
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Product of the elements cannot be 1

I have to show in a group $G$ of order $4k+2$ the product of the elements cannot be $1$. What I got so far is that there exists a subgroup $H$ of index $2$ in $G$ (considering left regular representation on $G$).Then $G/H$ is abelian. Now consider…
Via
  • 425
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Automorphisms of finite almost simple groups

Let $P$ be a finite nonabelian simple group. Let $G$ satisfy $$ P\leqslant G \leqslant {\rm Aut}(P), $$ where $P$ is identified with $\rm{Inn}(P)$. I am trying to see if $$ {\rm Aut}(G)\cong N_{{\rm Aut}(P)}(G) $$ Here is my attempt. Firstly, since…
Anvita
  • 633
3
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A Sylow $p$-subgroup can be a subset of union of the rest of the Sylow $p$-subgroups?

Let G be a finite group and assume that number of the Sylow $p$-subgroups is more than one. My question is this: "Can a Sylow $p$-subgroup be a subset of the union of the rest of the Sylow $p$-subgroups?" I think this is impossible. Every Sylow…
mesel
  • 14,862
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1 answer

How can I explain the sketch of the proof of classification of finite simple groups to someone who knows only basics of algebra?

This question was popped up in my mind when I read Finnish Wikipedia. How can I explain the sketch of the proof to layman? Is it worth to explain for example Ree groups in the text or just say something general like that kind of groups are important…
FinnWiki
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