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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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Subgroup order of $\mathtt{SmallGroup}(576,8661)$

I was studying the following problem : Groups of order $n^2$ that have no subgroup of order $n$ Is there any other divisor than 24 of $24^2$ such that SmallGroup(576,8661) has no subgroup of that order.
SSSM
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example where complements are not conjugate (in group theory)

In the Finite Group book by Isaacs, the concept of complement is defined as follows: Given the normal subgroup of $G$, a subgroup $H \subseteq G$ is an even complement $N$ in $G$ if $NH = G$ and $N \cap H = 1$. If $H$ complements $N$ in $G$,…
Sera
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Are $G/A$ and $G/B$ isomorphic if $A$ and $B$ have the same order?

Are $G/A$ and $G/B$ isomorphic if $A$ and $B$ have the same order? I think so but want to confirm. Let $G$ be finite.
CuriousKid7
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A question about normal sets in a finite subgroup

Let $G$ a finite non-abelian group and suppose that $N$ is a normal subset (not a subgroup) of $G$, that is, $aN=Na$ for every $a\in G$. Under what conditions $|N|$ divides $|G|$ ?
boaz
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Good Will Hunting Follow Up Question

I recently asked a question from the movie Good Will Hunting. Here is a link Good Will Hunting Problem Reasoning if you want more information. The matrix T is equal to the scalar z times the matrix shown below. I found the correct generating…
W. G.
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Permutation Cycle Question

If a=(1,4,3,2), then a(1)=1, a(2)=4, a(3)=3, and a(4)=2. Does a(5)=5 and a(6)=6 where these one cycles are neglected? Or does the permutation not exist for these values?
W. G.
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Group of order $210$

Is there a $G$ group of order $210$ satisfy following properties: for some $x,y \in G$, order of $x$ is $5$, order of $y$ is $3$ $x^4y^2=(yx)^6$. Notes: Problem is mine and there is no source. Easly we can show that $(xy)^7 = e$, (here $e$ is…
scarface
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Finite abelian groups isomorphic?

I know cyclic groups of the same order are always isomorphic, but as far as I'm aware finite abelian groups aren't necessarily cyclic. So is this statement true or false, and why?
Refnom95
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Generating set - inconsistency?

In my lecture notes $‹S›$ is defined as follows: Then later there is this: But surely this is exactly what $‹s,t›$ is? Directly from the Proposition, with $S=${$s,t$}, $H=${$s^jt^k$} with $s,t∈S$ and $j,k∈Z$ and then $H=‹S›=‹s,t›$. Surely then…
Refnom95
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Generator of a cyclic group among generators.

Let $G$ be a group generated by the finite set $X=\{x_1,\ldots, x_n\}$. Now suppose that $G$ is a finite cyclic group. It is clear that $G$ need not be generated by $x_i$ for any $i$. What additional hypothesis ensures that $G$ can be generated by…
user114539
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Double cover of an orthogonal group

Let $n \geq 3$, $(n, q) \neq (3,3)$ and $q$ be an odd prime power. It is known that $SO_{2n+1}(q)=\{A\in SL_{2n+1}(q): A^tJA=J\}$, where $J$ is a matrix with anti diagonal 1, 0 otherwise and $(SO_{2n+1}(q))'= B_n(q)$. Where can I find a complete…
Sara
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Show that all elements of a group are inversable

Let $E$ be a set , and let $f_a$ and $g_a$ be two functions such that for all $a \in E$ : $$f_a : E \rightarrow E$$ defined by $ f_a(x)=ax$, and $$g_a : E \rightarrow E$$ defined by $ g_a(x)=xa$. The Question is : Suppose that $f_a$ and $g_a$…
MathLearner
  • 91
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Normal subgroup $H$ and order of double coset $HgH$

Let $G$ a finite group and $H$ a sugroup of $G$. Show that $H$ is normal if and only if all the double cosets $HgH$ for every $g\in G$ have equal number of elements.
Roiner Segura Cubero
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Group Theory : Sum of order of elements of a group with finite and even order.

Consider a group $G$ which is finite and had even order. If we consider the sum of the order of the cyclic groups created by the elements of $G$, is this sum odd? How to prove this?
Almond Pie
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Subgroups of finite reflection groups

I am trying to understand finite reflection groups. Given a connected finite reflection group generated by $m$ reflections and let $S$ be a set of simple roots. Let $I \subset S$ be a subset of the simple roots. What is the type of the subgroup…
Jeff
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