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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3
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70 views

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 ...
0
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1answer
22 views

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 ...
2
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0answers
23 views

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: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
1
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1answer
38 views

Finite abelian p-group with an element of maximal order

I want to know, following theorem comes from which book? Theorem . Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there exists a subgroup $K⊆G$ such that: $ \langle ...
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2answers
81 views

Any group of order $15$ is abelian(without sylow theorem)

Prove that, any group of order $15$ is abelian (without help of Sylow's theorem or its application). What I have done so far is, by class equation we know that $|G|=|Z|+\sum\frac{|G|}{C(a_i)}$. Now ...
2
votes
1answer
32 views

Left adjoint to forgetful from modules to abelian groups

What is the left adjoint to the forgetful functor $U : R-\mathsf{Mod} \to \mathsf{Ab}$? Note here that $R$ is a general ring, not necessarily commutative. I've seen that they define it as $F A = R \...
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0answers
36 views

Prove that if $A,B,C$ are finite commutative groups and $A\times B\cong A\times C$ then $B\cong C$. [duplicate]

Prove that if $A,B,C$ are finite commutative groups and $A\times B\cong A\times C$ then $B\cong C$. Since $A,B,C$ are finite commutative groups hence we can write $A=\Bbb Z_{p_1^{\alpha_1}}\...
0
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2answers
27 views

Maximal subgroups of Z

In the ring of integers, the only maximal ideals are those generated by the prime elements. Is the same true for the group of integers? Are the only maximal subgroups of integers the ones generated by ...
0
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0answers
40 views

On conjugacy classes of involution

Let $G$ be a finite solvable group and $H$ be a normal subgroup of $G$. If all the involution of $H$ lie in a conjugacy class of $G$, then what can we say about the structure of $H$?
-1
votes
2answers
51 views

If K is a finite field, proof that $Gl_n(K)$ is not commutative [closed]

The following property was stated during a lecture in Algebra: If K is a finite field and $n \ge 2$ then $Gl_n(K)$ is a non-abelian finite group. I know how to proof that $Gl_n(K)$ is finite but,...
0
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2answers
48 views

Show that $o(g_1\cdot g_2)=o(g_1)\cdot o(g_2)$ [duplicate]

Let $G$ be an abelian group, $g_1,g_2\in G$ of finite order ($o(g_1)=m,o(g_2)=n)$ with $(o(g_1),o(g_2))=1$ (relatively prime). Show that $o(g_1\cdot g_2)=o(g_1)\cdot o(g_2)$. I have tried the ...
0
votes
3answers
160 views

Group Theory vs Graph Theory [closed]

I would like to know that, For a given graph can we find an associated finite group? If there are more than one such group, what are the differences and similarities between them? Here ...
2
votes
0answers
44 views

Isomorphic product of finite abelian groups

Suppose $X,Y,Z$ are finite abelian groups with $X \times Y \cong X \times Z$. How to show that $Y\cong Z$? If we assume that we can decompose $Y,Z$ into cyclic groups that are powers of primes, I ...
0
votes
1answer
60 views

Center of a group $G$, when the commutator subgroup has index 2 [closed]

Suppose that $G$ is a finite group, $M=G^{'}Z(G)$, $|\frac{G}{M}|=2$ and there is an element $x\in G$ such that $|C_G(x)|=4$. Is it true that $|Z(G)|>1$ ? ($G^{'}$ is the commutator subgroup and $Z(...
0
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3answers
37 views

A finite abelian group must contain an element which is the l.c.m. of the orders of its elements.

Let $G={g_1,...,g_n}$ be a finite abelian group of order $n$ and let $m =$ l.c.m.$(|g_1|,...,|g_n|)$. Since $G$ is finite (without loss of generality) suppose $g_1\cdots g_n = g_1$. We know $(g_1\...
1
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1answer
107 views

Abelian group in short exact sequence

If we have a short exact sequence of continuous group homomorphisms between abelian groups $$0 \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow X \rightarrow \mathbb{Z} \rightarrow 0,$$ can we ...
1
vote
1answer
38 views

Unique subgroup of index 2 in a finite abelian group.

Suppose $G$ is a finite abelian group, all elements of which are their own inverse. If the order of $G$ is greater than $2$, then prove or disprove that the subgroups of index $2$ in $G$ are not ...
0
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0answers
25 views

Property of Abelian Group

I am reading the Handbook of the Mathematical Logic and in the page 8 say: An abelian group $G$ has every element of order $\leq$ n if $G$ is a model of $\forall x [ x = 0 \vee 2x = 0 \vee \...
1
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2answers
69 views

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 ...
2
votes
0answers
40 views

Index of a maximal subgroup among normal abelian subgroups

Let $P$ be a $p$-group and $A$ maximal among abelian normal subgroups of $P$. Show that: 1) $A=C_P(A)$. 2) $|P:A|\mid (|A|-1)!$. 1) If $A$ is an abelian normal subgrup of a certain group $...
1
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1answer
101 views

Doubt about a kernel

I was reading the proof of Lemma 1.25 in this thesis and I thought I understood it, but I think I don't. The thing that I don't see clearly is in page 26 where he is showing that $\textrm{ker}\ \eta\...
3
votes
1answer
40 views

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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2answers
48 views

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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3answers
48 views

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 $R=\mathbb Z$, $M$ ...
1
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1answer
44 views

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 ...
0
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2answers
29 views

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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0answers
68 views

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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1answer
72 views

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 ...
2
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0answers
25 views

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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2answers
33 views

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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0answers
71 views

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 ...
3
votes
3answers
91 views

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 ...
4
votes
2answers
43 views

Isomorphism between $G$ and $\mathbb{Q}^{*}$

Let $\{G_{n}\}_{n\in \mathbb{N}}$ be a family of additive groups with $G_{1}=\mathbb{Z}_{2}$ and $G_{n}=\mathbb{Z}$ for $n\geq 2$ $$G=\bigoplus_{n\in \mathbb{N}}G_{n}$$ I want to prove that $G\cong \...
3
votes
1answer
62 views

Are all countable torsion-free abelian groups without elements of infinite height free?

The height of an element $g$ in an abelian group $G$ is the largest $n\in \mathbb{N}$ such that there exist an element $h\in G$ such that $n*h=g$. If $g$ has no such largest integer than $g$ is of ...
3
votes
3answers
65 views

A line avoiding an Algebraic group

Let $K$ be an algebraically closed field, and $G\subset (K,+)^3$ an algebraic subgroup (i.e. given as the zero sets of finitely many polynomial equations) of dimension 1. Is it clear that there is a ...
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0answers
45 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of $\...
2
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67 views

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. ...
1
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1answer
17 views

Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have $|f(...
4
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47 views

Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
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1answer
57 views

Quotient of direct sum of abelian groups [closed]

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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2answers
49 views

Showing a non-isomorphism of groups

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$. $\Bbb Z^*_n$ means integers up to $n$ coprime with $n$ I do not know how to do this. I have difficulties doing proofs ...
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2answers
35 views

Which one of the below options is correct?

I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$. Since order is $p^2$ thus $(Q)$ option is true. Can anyone suggest about option $(P)$? Thanks
0
votes
1answer
22 views

Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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1answer
41 views

Characteristic subgroups of order $2$

Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such ...
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26 views

Abelianizated free product of two groups

Given $$G=\mathbb{Z}_2*\mathbb{Z}_2=P(a,b\mid a^2,b^2)$$ among other things I wanted to show that this group is infinite, what I did is consider the words of the form $$abababa\ldots$$ they are all ...
7
votes
1answer
93 views

Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
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2answers
90 views

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ? ( I know that there 'is' a 'surjection' , but I don't know whether any surjective homomrophism from $\mathbb R$ ...
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0answers
25 views

freeness of vector spaces and abelian groups

This question is continuation of my previous question Extension of vector spaces and abelian groups Given a diagram of linear transformations of $K$ vector spaces $$B\xrightarrow{\epsilon} C\...
0
votes
1answer
69 views

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$ Can anyone provide references to places where this (or related) groups ...
2
votes
2answers
41 views

Extension of vector spaces and abelian groups

While reading about modules from Hilton & Stammbach's Homological algebra, I saw the following statement : $\Lambda$ is a ring. $\Lambda$ modules are generalizations of vector spaces and ...