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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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Question on Nilpotentcy of $G$ given $G/N$ nilpotent

If $G$ is a group and $N
hmmmm
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Order of elements within a group

If $G$ is a finite group of order (size) $n$ then, for any $g \in G$, the order (period) of $g$ is a divisor of $n$. Proof: $g$ must have finite order since $G$ is finite. If the order (period) of $g$ is $m$ then the order (size) of the cyclic…
user2850514
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Generated group of 2 element of order 2

Let $G$ be a finite group that is generated by $\alpha,\beta\in{G}$ of order 2, such that their product isn't of order 2. Show that $G$ is isomorphic to $D_n$ for some n.
Gyt
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Abstract Algebra group question

If $G$ be a finite group of $l$ elements. Suppose that $a$ belongs to $G$, and $\mathrm{ord}(a)=k$,can $k>l$? I think $k$ can't be bigger than $l$, because $k$ should equal $l$.
megan
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what can we know about this kind of group

Let G be a finite group,H is an arbitrary proper subgroup of G,H is solvable,but G is not solvable.then what can we know about group G?
user94782
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Are groups of component type always of Lie type, alternating or sporadic?

In http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups it was written that "A group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal…
FinnWiki
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Commutator and Frattini Subgroup

I'm traying to prove the following theorem : "Theorem: Let $G$ be a finite group, and $P \in \text{Syl}_p(G)$ where $p$ is the smallest prime dividing the order of $G$. Suppose that every subgroup of $P$ is normal in $G$, then $G \cong P \times A$…
Khaled
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Relating the three forms of the simple group of order 25920

There is a unique simple group of order 25920. It has three main forms that I am aware of $$ B_2(3) \cong C_2(3) \cong \; ^2A_3(2^2) $$ equivalently $$ O(5,3) \cong PSp(4,3) \cong PSU(4,2) $$ It is well known that $ B_2(q) \cong C_2(q) $ in other…
Ian Gershon Teixeira
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Order of automorphism group of an almost elementary abelian group

Let $A$ be an almost elementary abelian 2-group with an elementary abelian subgroup $B$ of index 2. If $|B|=2^m$ then what can we say about the order of automorphism group of $A$?
student
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Invariant factor decomposition, fundamental theorem of finite generated abelian groups

We know from the fundamental theorem that every finite group $G$ can be written as $G\cong Z_{n_1}\times\cdots\times Z_{n_t}$, where $n_i| n_{i+1}$. What is happening in the following example. $Z_6\cong Z_2\times Z_3$, but 2 does not divide 3.
ted
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What we know about sylow subgroups of $G/P$?

Let $G$ be a finite group and $P$ be it's normal $p$-subgroup. What we can say about the sylow subgroup of $G/P$? For example if we know that a $q$-sylow $(q\neq p)$ subgroup of $G/P$ is normal and non abelian what we can say about $q$-sylow…
Adeleh
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A group of order 2520

Let $G$ be a group of order 2520 and let $K$ be the maximal normal soluble subgroup of $G$. If we know that $G/K\cong A_5$, $K=C_2\times (C_7 :C_3)$, $C_G(K)=SL(2,5)$ and $G= C_G(K) K$, what would be the structue of $G$?( By $A_5$ I mean the…
Tina
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A condition for a finite group $G$ be nilpotent

Is true that a finite group $G$ is nilpotent if and only if $[x,y]=1$ for all $x,y \in G$, such that $(\mid x\mid, \mid y \mid) = 1$, where $[x,y] = x^{-1}y^{-1}xy$, ie, is the commutator of $x$ and $y$; and $(a,b)$ denotes the greatest common…
user59969
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Semidirect Product

Given a cyclic group $C_p$ and an abelian (noncyclic) group $K$ with $|K|$ divides $p-1$. Is it always possible to construct a nonabelian group $G \cong C_p\rtimes K$ with $Z(G) \cong C_p$?? If such group can be constructed, How could the…
Paul
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Computing the number of elements of order $2$ and $3$, in the groups $L_{3}(q)$

What are the number of elements of order $2$ and $3$ in the groups $L_{3}(q)$? Also let $r$ be a divisor of $q^2+q+1$. What is the nuber of elements of order $r$ in the groups $L_{3}(q)$?
Mark
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