# Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility.

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### Closure of the commutator group of Lie solvable group

Help me please to understand the following. Let $G$ be a connected Lie solvable group. Let $G':=\overline{[G,G]}$ be a closer of the commutator group $[G,G]$. Is it possible that $G'=G$? If so what if ...
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### A subgroup is equal to a group if it has elements of the conjugacy classes [duplicate]

Let $G$ be a group and $H$ be a subgroup of $G$. If $H$ contains at least one element of each conjugacy classes of $G$, then it holds that $H = G$? What would be the proof if this is true?
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### Intersection of two subgroups with order 39 and 65 is cyclic

Following is Question 9 of Chapter 10 of Abstract Algebra by Dan Saracino: Suppose $G$ is a group and $H$ and $K$ are subgroups of $G$ such that $|H|=39$ and $|K|=65$. Prove that $H \cap K$ is cyclic. ...
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### Groups and Cosets

I am studying the proof of the third Sylow's Theorem and I dont get this: Let $G$ be a finite group, $H$ a $p$-Sylow subgroup of $G$, $N(H)= \{g \in G : gH=Hg \}$. Note that $G = \cup_{g \in G}AgA$ (...
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### Let $\phi : G\to G'$ be a homomorphism and let $S'\subseteq G'$. Is $\phi^{-1}(\langle S'\rangle ) = \langle \phi^{-1}(S')\rangle$?

I've shown that $\phi^{-1}(\langle S'\rangle ) \supseteq \langle \phi^{-1}(S')\rangle$ and was wondering whether the other inclusion holds, although I've not been able to prove it nor find a ...
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### Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem

$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
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### How to prove the restricted Lorentz Group is connected?

Many textbooks claim that $SO^{+}\left(1,n\right)$ is the identity component of $O\left(1,n\right)$. But how do we know that $SO^{+}\left(1,n\right)$ is connected itself?
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### Proof the follow exercise of group theory [closed]

let $a \in G$ where $G$ is finite. If $a^n$ has $m$ conjugates and $a$ has $k$ conjugates, the m|k.
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### Gorenstein's proof of the classification of solvable CN-groups

I am reading Gorenstein's Finite Groups. Chapter 14 is about CN-groups, a (finite) group where the centralizer of every non-identity element is nilpotent. Theorem 14.1.5 gives the classification of ...
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### Show that if $G$ is finite and $G$ acts on $X$ and it is transitive, then $1=\dfrac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|$ [closed]

I know you have to relate the $\sum_{g\in G} |\operatorname{Fix}(g)|$ with $\sum_{x\in X} |\operatorname{Stab}(g)|$ and use the orbit theorem and stabilizer but I get very confused.
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### Proving a different version of the Third Isomorphism Theorem for groups

Theorem: let $\phi_1:G_1\to G_2$ and $\phi_2:G_2\to G_3$ be surjective homomorphisms, then $$\frac{G_1/\ker \phi_1}{\ker(\phi_2\circ\phi_1)/\ker(\phi_1)}\cong \frac{G_1}{\ker(\phi_2\circ\phi_1)}.$$ ...
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### Extension of a representation

My question is related with this Representation abelian subgroup of an abelian group. I want to show that X extends to a representation of G, which I think it's obvious from G being defined by its ...
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### Class of finite groups where all noncentral conjugacy classes have the same size but three different character dimensions

I'm looking for a class of groups where there are only two possible conjugacy class sizes, say $1$ and $k$, but at least three distinct dimensions of the irreducible characters. I know the conjugacy ...
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### Difference between products and coproducts.

I am struggling with understanding the difference between products and coproducts in category theory. For the category of abelian groups what part of the universal property of coproduct implies the ...
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### Discrepancy between order of the identity element & its inclusion in n-torsion sets

The order of the identity element of a group is usually considered to be one because $1 * e = e$ or $e^1 = e$. However, the text on torsion points (from Mathematical Cryptography by Silverman, Pipher ...
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### How to conclude the order of $G$ is $p^{m+n}$.

I have this exact sequence $$0\to \mathbb{Z}_{p^m}\overset{f}\to G\overset{g}\to \mathbb{Z}_{p^n}{\to} 0$$ where $G$ is finitely generated. I want to prove that $|G|=p^{m+n}$ and a friend told me to ...
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### Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
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### Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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### Show that $\langle a\rangle$ is a normal subgroup of $\langle a,b\rangle$ where $a$ and $b$ are permutations

I'm studying for an exam and in a previous year we had the question: Let $a=(12345678)$, and $b=(26)(48)$ and $G=\langle a,b \rangle$. Show that $\langle a \rangle$ is a normal subgroup of $G$ and by ...
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### Permutation Groups: Find $x$ such that $x^5 = (12345)$
I am wondering about how to solve question 35 from chapter 5 (Permutation Groups) from the 10th edition of Gallian’s Abstract Algebra. The full question is as follows: What is the smallest $n$ for ...