# 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.

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### Regular normal subgroups.

Hi: I don't understand the proof ($G$ is a finite group). From the equality I get $\beta=\alpha^{{(x_{\beta})^g g^{-1}}}$. Unfortunately the exponent does not necessarily belong to $N$ and so I can't ...
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### Proving that a group is abelian!

Supposing we have a group $(G,*)$ (Which we don't know the elements of). It is given that for each $x, y \in G$, exists $x*y*x=y$. That's the only information we receive, How can we prove that group G ...
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### How the square model of Fano 3-space, a.k.a. $PG(3, 2)$, embeds in the MOG?

I'm studying deeply the Curtis' MOG original article, i.e. a New combinatorial approach to $M_{24}$ and I'm still struggling a bit in reading the final 35 6x4 matrices representing the MOG. Actually ...
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### An abelian group $G$ that has order 216, but $6G$ has order 6

I’m trying to prove the following statement: “if G is a finite abelian group with $|G|=216$ and $|6G|=6$, then determine $G$ up to isomorphism.” I know that by the fundamental theorem of finite ...
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### Let $H\le G$ of index $3$. Prove that either $H\unlhd G$, or that $H$ has a subgroup $N$ of index $2$ in $H$ such that $N\unlhd G$.

Let $H$ be a subgroup of a group $G$ of index $3$. Prove that either $H$ is normal, or that $H$ has a subgroup $N$ of index $2$ in $H$ such that $N$ is normal in $G$. All I could show is that if $H$ ...
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### Cyclic groups as normal sobgroups of SO(3)

I have to argue why none of the cyclic groups $C_n$ is a normal subgroup of $SO(3)$. Nevertheless, I haven't found an argument for that. First I used the definition of normal subgroup: If some $C_n$ ...
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### Kummer Theory: Understanding the isomorphism $F^\times \cap E^{\times n}/F^{\times n} \to \operatorname{Hom}(G,\mu_n)$

Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$. Now I am trying to understand the following section from Milne's Fields and Galois Theory (page 73): The part "...
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### Understanding the map about the classification of all abelian extensions with Galois groups with a fixed exponent (Kummer Theory)

Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$. Also, let $E/F$ be a finite Galois extension with Galois group $G$. Now I am trying to understand the following Theorem ...
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### Order of $(1 \,3)(2 \, 5 \, 4)$ in $S_5$

What is the order of $(1 \,3)(2 \, 5 \, 4)$ in $S_5$? From number theory, I remember that we defined the order to be the smallest positive integer $k$ for which $a^k \equiv 1 \pmod{n}$ and also $a$ ...
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### Interleaving a fixed element into a given sequence of elements of a permutation group and the image of a point

Let $G \le \operatorname{Sym}(n)$ be a permutation group on $\{1,\ldots,n\}$. For $\alpha \in \{1,\ldots, n\}$ and $x \in G$, write $\alpha^x$ for the application of $x$ to $\alpha$. Fix some $g \in G$...
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### The proof of $S_n \cong A_n \rtimes \{e, (12) \}$

$\blacksquare~$ Problem: Let $G = S_n, H = A_n$ and $K = \{ e, (12) \}$. Show that $S_n \cong A_n \rtimes K$. $\blacksquare~$ My Approach: Let $G = S_{n}$. Where $S_{n}$ is the symmetic group of ...
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### A family of groups as a monoidal category

1.Context My lecture notes present the following example of a monoidal category: Let $G:=(G_n)_{n\in \mathbb {N_0}}$ be a family of groups with $G_0$ the trivial group with one element. We define a ...
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### How to prove that pqr order group is solvable group？ [duplicate]

Let $G$ be a group and $|G|=pqr$, where $p,q,r$ are prime numbers that are not necessarily distinct. Show that $G$ is solvable. I try to discuss the classification of groups of order $pqr$ and I also ...
How does one perform cycle multiplication? It seems that every textbook I read has a different notation for this and it's not clear at all. Suppose I have $(123)(134)$ now some book stated that the ...
My question is taken out of a proof in the book "Infinite Soluble Groups" by Robinson and Lennox. Let me paste the proof first: Some reminders: a soluble group $G$ has FATR (finite abelian ...