# Tagged Questions

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 studies groups.

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### Write cyclic groups of order $p^n$ in terms of simple groups

Some say that studying simple groups helps you understand the structure of non-simple groups. How can I write in terms of simple groups $\mathbb{Z}_{p^n}$? Eg. $\mathbb{Z}_9$
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### Abelianization of the free product of two cyclic groups.

Suppose that $G=G_1*G_2$ where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$ respectively. Show that $G/[G,G]$ has order $mn$. Can anyone help me?
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### A field having an automorphism of order 2

The following fact is used in the Unitary space. If $F$ is a field having an automorphism $\alpha$ of order 2. Let $F_0=\{a\in F: \alpha(a)=a\}$. Then $|F:F_0|=2$. Is there any easy proof (or ...
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### Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
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### Subgroups of $\mathbf{Z}/20\mathbf{Z}$ using Lagrange's Theorem

According to Lagrange's Theorem, what are the possible sizes of the subgroups of $\mathbf{Z}/20\mathbf{Z}$? I have no idea how to go about answering this. I have a feeling that I should be ...
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### How can i prove that a cartesian product is isomorphic to another cartesian product

$\def\<#1>{\left<#1\right>}\def\Z{\mathbb Z}\<\Z_6, \oplus> \times \<\Z_{10},\oplus>$ is isomorphic to $\<\Z_2, \oplus> \times \<\Z_{30}, \oplus>$ i know i have to ...
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### Topological group with discrete topology

Let $G$ be a topological group. I came to know that if I can show the existence of a homeomorphism of $G$ which moves only finitely many points of $G$, then $G$ has only discrete topology. How can I ...
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### Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$ . . .

Let $U_g$ be the group of units of $\mathbb{Z}/g\mathbb{Z}$. Then $U_g$ is a subgroup of itself. For every unit $c$ of $U_g$, show the coset, $cU_g = U_g$. Show that the product of the elements of ...
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### How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
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### Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
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### I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
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### Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
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### “finite part” of an abelian pro-p group

I'm trying to understand the proof of the following: Let $G$ be an abelian pro-$p$ group, and let $N\leq _O G$ be an open subgroup such that $N\cong\mathbb{Z}_p$. Then $G\cong\mathbb{Z}_p\times T$ ...
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### Quaternion Group as Permutation Group

I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it. So I googled to see if somebody else had put the ...
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### Does this subgroup of $\mathrm{SL}(2,\mathbb{C})$ have a a name?

The set of matrices $g$ characterized by $g=\begin{pmatrix}a&ib\\ ic&d\end{pmatrix}$, where $a,b,c,d \in \mathbb{R}$ and $ad+bc=1$, can be easily shown to be a subgroup of ...
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### Are elements commutative when subgroups are?

Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
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### Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
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### A normal subgroup problem

Let $G$ be a group in which, for some integer $n>1$, $(ab)^{n}=a^{n}b^{n}$ for all $a,b \in G$. Show that $G^{(n)}=\{x^{n} \mid x \in G\}$ is a normal subgroup of $G$. $G$ could be easily ...
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### semidirect product of cyclic and p-groups

How does conjugacy classes subgroups of a semi direct product group cyclic group of order (q-1) where q is a power of p, look like?
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### why is it that the conjugate of a+bi is a-bi?

if a+bi is an element of a group, then its conjugate is a-bi, how can we prove this by using the fact that the conjugate of an element g of a group is h if there is an x in the group such that ...
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### Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
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### Conjugate subgroups of $GL_n(K)$

Let $K \subset L$ two fields, $G$ and $G'$ subgroups of $\mathrm{GL}_n(K)$. Assume that $G$ and $G'$ are conjugate in $\mathrm{GL}_n(L)$. Are $G$ and $G'$ conjugate in $\mathrm{GL}_n(K)$? ...
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### Properties about order of a general group

For a group, $G$, is it true that $o(Z(G))\cdot o([G,G]) \leq o(G)$ where $Z(G)$ denotes the centre of $G$ and $[G,G]$ denotes the commutator subgroup of $G$?
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### Prove that any group of order 15 is cyclic. [duplicate]

Prove that any group of order $15$ is cyclic. I know that if order of group is a prime then the group is cyclic, but how to approach such questions?
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### Pairs of $2\times 2$ matrices generating free groups.

The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, ...
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### Is it “group axiom” or “group definition”?

Some text books of group theory use "group definitions" when introducing group, and some other text books use "group axioms". But it is obvious that terms "definition" and "axiom" are different. Which ...
### Show for each $c$, $\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$ is an abelian group under multiplication of congruence classes
Show for each $c$, the set $$\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$$ is an abelian group under multiplication of congruence classes.