# 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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### Algebraic Structures that do not respect isomorphism

One of the first things a student learn in Algebra is isomorphism, and it seems many objects in algebra are defined up to isomorphism. It then comes as a mild shock (at least to me) that quotient ...
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### Commensurability for vector spaces

Let me start by saying I am a not a mathematician and I have not studied group theory (just a few brushes here and there) but after reading I have a very basic understanding of commensurability as ...
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### Order of factor group

Question: Determine the order of $(\mathbb{Z} \times \mathbb{Z})/ \left<(4,2)\right>$. Is the group cyclic? I want to first apologize for the way this post is written. I'm on the road and ...
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### Determining a group given some elements

Say we have a group G and we know some of the elements (but not all). How does one determine the order and list all the elements of the group in an intuitive way? In this case G is the smallest ...
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### For a prime $p$ if $p^m = p^n+2\cdot p^k$ then $p=3$.

I read an article on commuting graphs of groups and at some point, author gets the equality $|\langle x,Z\rangle| = |Z|\cup 2\cdot |x^G|$ where $Z$ denotes the center of the $p$-group $G$ and $x^G$ ...
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### Finite number of orbit type for compact Lie group actions

For the linear SO(3) actions, the number of orbit types (or the number of isotropy class) is finite : this seem to be a classical result coming from Bredon (Introduction to compact transformation ...
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### Pre-image of a join of subgroups

Let $\phi : G \longrightarrow H$ be a surjective homomorphism. Suppose that $W\leq H$ and let $x\in H$ such that $x= \phi(g)$ for some $g\in G$. I would think that the following relation holds, though ...
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### Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit?

Does $\text{Gal}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ act on $\overline{\mathbb{Q}}$ with an infinite orbit? I know the definition of orbit and I know that $\sigma$ in Galois group changes roots. I ...
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### How to find the smallest set of generating elements in a group?

Is there a systematic procedure for finding the smallest set of generating elements of a finite group?
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### Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I$ and $detX=1\}$...
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### Counting subgroups of free product of copies of $\mathbb{Z}$ with certain index

For a natural number $n$, let $Z_n=\mathbb{Z} \ast \cdots \ast \mathbb{Z}$ denote the free product of $n$ copies of the integers. Let $m$ be a further integer. $\textit{Question:}$ Is there a ...
Suppose $G$ is a group and $T$ is an automorphism of $G$ of order $k$. We create a group $\{G,T\}$ (Construction given in Herstein 2nd edition Pg 69). Now in the explanation given in text, it ...
Let $\phi : G \longrightarrow H$ be an epimorphism. If $W$ and $K$ are cojugate in $H$ then $\phi^{-1}(W)$ and $\phi^{-1}(K)$ are conjugate in $G$. Firstly is this true for the above condtions? I ...