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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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Two subgroups whose orders are greater than square root of the group order have no trivial intersection

Two subgroups whose orders are greater than square root of the group order have no trivial intersection I cannot come up with a critical idea.
user233454
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The Glauberman -Thompson theorem

What is the Glauberman-Thompson theorem? I read in a paper, if $% N_{N}(Z(J(P))$ is nilpotent (where $P$ is a Sylow $p$-subgroup of $N$ and $N$ is a minimal normal subgroup of $G$), then by the Glauberman-Thompson theorem, $N$ is $p$-nilpotent.
User1257
  • 43
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finite soluble groups and Hall subgroups

It is rather obvious that every Hall π-subgroup is a Sylow π-subgroup. In general, however, G need not contain any Hall π-subgroups. For example, a Hall {3,5}-subgroup of $A_5$ would have index 4, but $A_5$ have no such subgroups. (Why?)
ghazal
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Groups of order 36 - another step in lemma 5.4.

This is a follow up to my question last night Groups of order 36 where I was confused about the first step of Lemma 5.4 of http://matwbn.icm.edu.pl/ksiazki/fm/fm92/fm9211.pdf. I am now confused about one of the last steps - that R cannot be…
Hari Rau-Murthy
  • 1,346
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prove that $O^{P^{'}}(G)=P^{G}=P[P,G]$.

suppose that $G$ is finite group and $p$ is prime number.prove that if $P$ is a $p$-sylow subgroup of $G$ then $O^{p^{'}}(G)=P^{G}=P[P,G]$ which $P^{G}$ is normal closure of $P$ in $G$ . any hint or guidance or references to study will be…
kpax
  • 2,911
-1
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Subgroups in $G$ of the form $gHg^{-1}$

Let $G$ be a group and $H$ be a subgroup of finite index. Prove that there is only a finite number of distinct subgroups in $G$ of the form $gHg^{-1}$ where $g$ belongs to $G$.
Zonash ali
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show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of 2.

show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of $2$. please help me how should start and proceed,any guide or hint will be great,thanks.
kpax
  • 2,911
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A question about $PSL(2,8)$

Can anybody tell me how to construct the character table of $PSL(2,8)$? I need a specific method.
Bruce Thou
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1 answer

$G = \mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k}$ contains a element of order $m$ iff $m\mid n_1$.

Let $G$ be a finite Abelian group: $$G=\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k}$$ and $n_k \mid n_{k-1} \mid \cdots \mid n_1$. Show that $G$ contains an element of order $m$ iff $m\mid n_1$.
Muniain
  • 1,453
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group theory of finite abelian groups

Let $G$ be a finite abelian group of order $p^nm$ , where $p$ is prime that does not divide $m$. Then show that $G = H\times K$ , where $H = \{x\in G | x^{p^n} = e \}$ and $K= \{x\in G | x^m= e\}$.
RABINDRA MOHARANA
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Prove that $\langle x, y\mid x^8=y^2=1, \:y^{-1} xy = x^3\rangle$ is order 16?

I want to solve above question by presentation free group theory. But I don't know how I should solve that. Could you possible help me?
Smith
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problem on group theory

Prove that a group with more than one element contains an element of prime order.
MAINAK BHOWMIK
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