# Tagged Questions

Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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### Show that $G$ is Abelian if and only if $f: G\times G \to G$ is a homomorphism.

Let $G$ be a group. Let $H$=$G\times G$ be the direct product of $G$ with itself. Define $f: H\to G$ to be $f((g,h))=gh$ for any $(g,h)\in H$. Show that $G$ is Abelian if and only if $f$ is a ...
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### Let A be a finite group and P be a normal p sylow subgroup. What is the connection between P and $Tor_p(A)$

Let A be a finite group and P be a normal p sylow subgroup. can there be an element $g \in A$ where $order(g) = p^x$ where x>0 and $g \notin P$ ? what I really try to understand is the connection ...
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### What triple “tensor product” is this? Is it just isomorphic to a double tensor product?

Consider the abelian groups $A = \Bbb{Q}^{\times}, B = \Bbb{Q}^{\times}, C = \Bbb{Z}^+$. What if we formed a product like: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
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### What is $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$ isomorphic to?

This question came up after I'd solved the following exercise: Determine the order of $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$. Is the group cyclic? I had no trouble solving the ...
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### What is $\mathbb Z^2/\text{Im}(\phi)$ isomorphic to in the following case?

Let $\phi:\mathbb Z^2\to\mathbb Z^2$ be the map $(x,y)\mapsto (x+y,2y)$. I need to find $\mathbb Z^2/\text{Im}(\phi)$. My guess is that this is isomorphic to $\mathbb Z_2$ but I am having ...
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### The center of a group is an abelian subgroup

Let $(G,\circ)$ be a group and let $Z(G):=\{x \in G : ax=xa \ \forall \ a \in G\}$ be the center of $G$. How can I show that $Z(G)$ is an abelian subgroup of $G$? What I did so far: $Z(G)$ is a ...
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### Submodules of a module with a given property

I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module. If $M$ is a finitely generated are the submodules of $M$ finitely generated? If $R=\mathbb Z$, $M$ ...
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### Number of Nonisomorphic Subgroups of Finite Abelian Group

Lets say I have an abelian group $G$ with order $n$ and I am given the primary components of $G$ and their type. How can I determine how many nonisomorphic subgroups of $G$ there are? And as an ...
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### Let G be a finite Abelian group of order $p^nm$, where p is a prime that does not divide m. Then $G=H\times K$ where H and K are the following sets.

I'm trying to follow this proof in my textbook, Contemporary Abstract Algebra by Gallian (p231) but I'm having trouble understanding what's going on. He writes Let G be a finite Abelian group ...
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### Questions about completion chapter in Atiyah-Macdonald

I was reading the completion chapter of Atiyah-Macdonald. I have the following questions: (i) What is the topology in the completion group of the topological abelian group? I saw an answer here. But ...
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### How do I prove this seemingly obvious property of subgroups

The statement is the following: Given an abelian group $G=\langle a_1,...,a_t\rangle$, and a subgroup $H$ of $G$, we need at most $t$ elements to generate $H$; i.e. $H=\langle b_1,...,b_t\rangle$ for ...
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### Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$.

Describe all the one-dimensional complex representations of a finite abelian group $G$. Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$. attempt: ...
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### Abstract Algebra Elementary Properties of Groups

This is Excercise 4.A.5 from Pinter's "A Book of Abstract Algebra": Let $a$, and $x$ be elements of a group $G$. Solve for $x$ in terms of $a$. Solve Simultaneously: $x^2 = a^2$ and $x^5 = e$ ...
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### When are groups subgroups of a same group?

Assume that $G_{1}$ and $G_{2}$ are groups and that $G_{1} \cap G_{2}$ has a group structure that makes it a common subgroup of $G_{1}$ and $G_{2}$. In other words, the set $G_{1} \cap G_{2}$ is a ...
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### If $G = \{g_1, g_2, …, g_n\}$ is a finite abelian group, then for any $x \in G$, $xg_1 \cdot xg_2 \cdots xg_n = g_1 \cdot g_2 \cdots g_n$

Let $G = \{g_1, g_2, \dots, g_n\}$ be a finite abelian group, prove that for any $x ∈ G$, the product $$xg_1 \cdot xg_2 \cdot \cdot \cdot xg_n = g_1 \cdot g_2 \cdot \cdot \cdot g_n.$$ I can easily ...
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### On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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