Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

33,582 questions
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Looking for group $G$ such that $G$ nilpotent, $N \leq G$, $N \cap Z(G) = \{1\}$

$G$ nilpotent, $N \leq G$, $N \cap Z(G) = \{1\}$. Note that it is a common exercise to show that if we add the assumption t hat $N \triangleleft G$ then $N \cap Z(G) \neq 1$. So I'm really looking for ...
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Why only define word only with finite product of elements?

My textbook says that the definition of a word is $(s_1,s_2, s_3 , \dots )$ where $s_i\in S\cup S^{-1}$ and $s_i =1 \;$ for all $i$ sufficiently large. I don't get that why is this "$s_i =1 \;$ ...
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Abelianization is left adjoint to the forgetful functor

I am trying to show that the map $ab:Grp \rightarrow AbGrp$ is left adjoint to the forgetful functor. I have seen in a similar question the approach is as follows: Let $f:G\rightarrow H$ be a group ...
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Intersection of Normal closure and Center

Let $G$ be an HNN-extension $\langle a,t\mid t^{-1}a^2t=a^2\rangle$. Then I am going to show that the normal closure of $t$ intersects the center of $G$ trivially. In the first step by using normal ...
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Does there exist some sort of classification of horny groups?

It is well known, that any finite group of order $n$ is isomorphic to a subgroup of $S_n$. Let’s call a finite group $G$ horny iff it is not isomorphic to any subgroup of $S_{|G|-1}$ . Does there ...
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Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
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Dihedral group generated by $\langle r,s\rangle$ for all $n$

Under wikipedia for Dihedral groups it claims the following: The $2n$ elements in $D_n$ can be written as $\{e,r,r^2,r^3,\ldots,r^{n-1},s,rs,r^2s,\ldots,r^{n-1}s\}$. I know why this is true and it ...