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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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An Equivalence class in a group of permutation

We know that every equivalence relation, induced by a partition on a set for example $X$ , make some equivalence classes. Now, if a group $G$ acts on $X$ then the associated equivalence classes are exactly the orbits of $X$ under the action of…
Mikasa
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Induced automorphisms of regular automorphisms on factor groups

If you have a $\phi$-invariant, normal subgroup $N$ (so $\phi(N)=N$) of a finite group $G$, for an $\phi$, then you get an induced automorphism of $G/N$ by $gN\mapsto \phi(gN)=\phi(g)N$. The order of the induced automorphism is a divisor of…
Ske
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Do "finitely generated abelian group" and "finite abelian group" mean the same thing?

I'm a little confused. In my textbook, they try to determine the abelian groups of order $1500 = 2^{2}\cdot 3\cdot 5^{3}$. They find the following families of elementary divisors: $\{2,2,3,5^{3}\}$ , $\{2,2,3,5,5^{2}\}$, $\{2,2,3,5,5,5\}$,…
bemyguest
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Not isomorphic groups.

I have to prove that $S_{4}$ is not isomorphic to $C_{2}\times A_{4}$ and I have no idea to do it. Some ideas? I have tried a lot of methods but with no luck.
bemyguest
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Groups of order 36

Prove: If a group $G$, of order 36 has a subgroup of order 18 ,$H$, then $G$ either has a normal subgroup of order 9, or a normal subgroup of order 4. This came about while reading the same article asked about in Group of order 36 stackexchange. …
Hari Rau-Murthy
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if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable.

if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable. we have that $G$ is solvable,I want to show that all factors of derived series are cyclic. but no achievement,so it will be great to help me with…
kpax
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Maximal Subgroups in Groups of Order $p^k$

The following question is from a past problem set in a course on group theory. For reference, the text used is by Derek Robinson, entitled "A Course in the Theory of Groups". "Show that in a group $G$ whose order is some power of a prime $p$, $H$ is…
inkievoyd
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Is there a special name for this single element discrete subgroup of Mobius group

Is there a special name for this discrete subgroup of Mobius group with single element: $$A=\begin{bmatrix}0 & i\\-i & 0\end{bmatrix}$$ and $\det A=-1$ and $A^2=I$. Thanks- mike EDIT: $$A=-\sigma_y=-\begin{bmatrix}0 & -i\\i & 0\end{bmatrix}$$ where…
mike
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suppose $G$ is a finite group, $G$ is nilpotent iff every quotient group of $G$ has no trivial center.

suppose $G$ is a finite group, $G$ is nilpotent iff every quotient group of $G$ has no trivial center. any hint or guidance will be great,I badly stuck in this one,thanks.
kpax
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every principal factor of a finite soluble group is elementary abelian.

every principal factor of a finite soluble group is elementary abelian. I am a little confused in a lot of definitions and I stuck in this exercise,this is so great if you just give me hints that I could be in right way,I really appreciate,thank…
kpax
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suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2.

suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2. it will be great if you help me how should I prove this.any note or reference will be great.thank you so much.
kpax
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Finding Sylow p subgroups and determining group structure using Sylow's Theorem.

Find the number $n_p$ of Sylow $p$ subgroups of $Alt(5)$ for $p=5,3,2$. So $Alt(5)=60=3*4*5$ Then for $p=5$, $n_5$ $=2,3,4,6,12$ but only $6=1$ mod $5$, hence there are $6$ Sylow $5$ subgroups in $Alt(5)$ Then for $p=3$, $n_3$ $=2,4,5,10,20$ but…
ZZS14
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Why are the composition factors of Sym(4) and Sym(5) unique?

I have worked out the CS and CF, and I have been given in my notes that they are unique, but no explanation as to why and in a past paper I am asked for reasoning. Is it because Alt(4) and Alt(5) are the options for maximal groups of Sym 4 and Sym 5…
ZZS14
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Show that G', the commutator subgroup of G, is normal in G. Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G.

Show that G', the commutator subgroup of G, is normal in G. Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G. So the definition of the commutator subgroup is that; In the group G, let G' be the subgroup generated by the…
ZZS14
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Identify a semidirect product $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$

I'm studying for the first time semidirect product and I'm trying to learn how to identify some of them. For example $\mathbb{Z}/4\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$ I red that, for $H\rtimes K$ consists to identify all the homomorphisms…
MaríaCC
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