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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2
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2answers
169 views

The sum of the orders of all elements of a group G

Let $Z$ be a finite group and denote $k(Z)$ the sum of the orders of all the elements of the group $Z$. I have to determine min $k(Z)$ and max $k(Z)$ when $G$ goes through the set of the abelians ...
5
votes
0answers
47 views

Groups and Rings [duplicate]

Is every abelian group the additive group of some ring? I would very much appreciate if someone could show me if this is false or true, something I'm thinking about and finding hard to prove so im ...
0
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1answer
81 views

Is there a Relationship between Quantum Groups and Lie Groups?

I know that the Lie Group is all about continuous transformation groups. I know that the quantum group denotes various kinds of noncommutative algebra with additional structure. Transformation group ...
7
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2answers
134 views

Divisible groups, exercise from Rotman's theory of groups

The following exercise is from Rotman, An Introduction to the theory of groups, 4th ed, p324. "The following conditions on a group G are equivalent: (i) G is divisible, (ii) Every nonzero quotient of ...
5
votes
1answer
74 views

Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?

Let $G=G_1\times G_2$ be a direct product, and let $H\triangleleft G$ be a normal subgroup such that $H\cap G_1=H\cap G_2=\{1\}.$ Then $H$ is abelian. I considered the commutators of two elements ...
1
vote
1answer
158 views

Non-abelian group in which $\forall_{a,b\in G} (ab)^3=a^3b^3$ [duplicate]

Give an example of a non-abelian group, in which $(ab)^3=a^3b^3$ for every element $a,b$ in $G$. I understand that such a group should be of order divisible by 3 (see Problem from Herstein on group ...
1
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0answers
340 views

Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence \begin{equation} 0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0. \end{equation} ...
3
votes
3answers
173 views

Isomorphisms of direct products of finite abelian groups

Suppose $G_1, G_2, H_1, H_2$ are finite abelian groups with $G_1 \times G_2 \cong H_1 \times H_2$, and $G_1 \cong H_1$. Prove that $G_2 \cong H_2$. Since the groups are finite, the isomorphisms ...
2
votes
1answer
63 views

Abstract Algebra Subgroup Help

Suppose that $G$ is an additive abelian group. Show that $H = \{a \in G\,|\, a + a = 2a = 0\}$ is a subgroup of $G$. Proof: (1) Nonempty Now $e = a + a = 2a = 0 \in H$, so $H$ is nonempty. (2) ...
0
votes
1answer
85 views

Show that $G$ has exactly one subgroup of order $8$.

I have this problem: Let $G$ be an abelian group of order $72$. Show that $G$ has exactly one subgroup of order $8$. I've seen how to find all abelian groups (up to isomorphism) of order $n$, ...
1
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1answer
134 views

Subgroup of an abelian Group

I think I have the proof correct, but my group theory is not that strong yet. If there is anything I am missing I would appreciate you pointing it out. Let $G$ be an abelian group (s.t. $gh = hg$ $\...
1
vote
2answers
126 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
1
vote
1answer
33 views

$H_{0}(A)$ in chains with zeros?

If we have a non-zero abelian group A and $0\rightarrow{A}\rightarrow0$, am I correct in thinking $H_{0}(A)=A$? If so why because I'm a bit confused and to the image & kernel in this case...
0
votes
2answers
53 views

Show that given group is abelian

There's a set consisting of 2 elements: G = {a,b}. In this set we define an operation * in the following way: $$a*a=b*b=a$$ $$a*b=b*a=b$$ The question says: "Show that (G, *) is a commutative group"...
1
vote
1answer
139 views

Does there exist an abelian group that can be made into $\mathbb{Q}$-module in more than one way?

Let $X$ and $Y$ denote $\mathbb{Z}$-modules. Then if $X$ and $Y$ have equal underlying abelian groups, we may deduce that $X=Y$. Is this still true if we replace $\mathbb{Z}$ with $\mathbb{Q}$? ...
1
vote
1answer
68 views

Under what conditions should a sub-group of a direct sum, itself be a direct sum?

This is a question I'm struggling a couple of days with: Let $G_1,G_2$ be abelian groups, and let $H$ be a subgroup of $G:=G_1\oplus G_2$. Under what conditions must $H$ be a group of the form $H_1\...
3
votes
3answers
87 views

Necessary and sufficient conditions for $H$ to be abelian, given a homomorphism from an abelian $G$ into $H$

It's trivial to show that if $ G\cong H$, then $G$ is abelian iff $H$ is abelian. However (Possibly trivial as well :), given that $G$ is abelian and there exists a homomrphism $\varphi:G \rightarrow ...
5
votes
1answer
98 views

Prove a result on the size of the minimal set that generates a finite abelian group

I am asked to prove the following: Let $G$ be a non-trivial, finite abelian group. Let $s$ be the smallest positive integer such that $G = \langle a_1,...,a_s\rangle$ for some $a_1,...,a_s \in G$. ...
6
votes
2answers
977 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does "$G$ is abelian" or "$G$ is cyclic" imply "$\text{Aut }G$ is cyclic"?
0
votes
1answer
25 views

Find an abelian subgroup of $GL(2n, q)$ of special order

I want to prove that $GL(2n, q)$ where $q$ is even, has an abelian subgroup of order $q^{2n - 1}$. please help me.
0
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0answers
117 views

A question regarding the Fundamental Theorem of Finitely Generated Abelian Groups.

The Fundamental Theorem of Finitely Generated Abelian Groups states Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form $$\Bbb{Z_{(p_1)^{r_1}}}...
1
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1answer
93 views

Abelian group and number of solutions of $x^n=e$

Let $G$ be an abelian group in which the number of solutions of $x^n=e$ is at most $n$ for every positive integer $n$. Show that $G$ is cyclic. I've shown this nice fact only when $|G|$ is finite. ...
0
votes
2answers
54 views

In the group A/B find the order of the coset (x$_1$+2x$_3$)+B

A is an abelian free group, with the base x$_1$,x$_2$,x$_3$. will be B a sub-group that created with x$_1$+x$_2$+4x$_3$,2x$_1$-x$_2$+2x$_3$. In the group A/B find the order of the coset (x$_1$+2x$_3$...
4
votes
1answer
171 views

If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian.

Let $G$ be a finite group which possesses an automorphism $σ$ such that $σ(g)=g$ if and only if $g=1$. If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian. This is what I got ...
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2answers
146 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all $...
2
votes
2answers
38 views

Ordered abelian groups

Consider the following axioms: 1) $\ x+(y+z)=(x+y)+z$ ; $\forall x \forall y \forall z$ 2) $\ x+0=x$ ; $\forall x$ 3) $\forall x$ $ \exists y$ such that $\ x+y=0$ 4) $ \ x+y=y+x$ ;...
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5answers
190 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
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1answer
58 views

Two free basis of a free abelian group

We have a free abelian group $A(X)$, where $X$ is its free basis, and let $Y$ another free basis for $A(X)$. We know that every $g\in A(X)$ can be expressed as $g=a_1x_1+...+a_nx_n$ where the $a_i$'s ...
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2answers
239 views

Unsatisfying proof on project crazy project (finite group is not divisible)

I found this proof that no finite group is divisible.(here) Let A be a finite divisible group then there are elements $x_k$ such that $x_k^k=1$ for each natural k. Then that would mean there is an ...
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2answers
78 views

Abstract algebra question: abelian group.

$H=\left\{\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix} : b \in\mathbb{R}\right\}$ $G=\left\{\begin{pmatrix}a & b \\ 0 & d\end{pmatrix}: a, b, d \in\mathbb{R}, ad\ne0\right\}$ $H$ is ...
2
votes
4answers
424 views

Why is $\operatorname{Hom}(A, B)$ an abelian group?

Can someone please explain why a Hom-set (the set of all morphisms between two abelian groups $A$ and $B$) does also form an abelian group with addition? By the way both groups $A$ and $B$ have the ...
3
votes
1answer
156 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
1
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1answer
82 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
1
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1answer
75 views

Proof of Abelian group

I have run into something I don't understand - a proof that group $P= \{a+b\sqrt5: a,b \in \Bbb Q\}$ is abelian considering usual addition operator $+$. Authors state that checking if the difference ...
2
votes
1answer
636 views

Prove that if a group is nilpotent , then its quotient with its Frattini subgroup is abelian

I know that : 1) Nilpotent group is solvable. 2) Subgroup of a solvable group is solvable. 3) Solvable and simple group is abelian. Now I should use these facts to prove it.
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2answers
601 views

Prove that the only homomorphism between a simple non-abelian group G and abelian group A is trivial

Prove that the only homomorphism between a simple non-abelian group $G$ and abelian group $A$ is trivial. OK. So G is a perfect group (G' = G) and A is abelian (A' = {1})
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votes
4answers
257 views

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian.

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian. I can prove that if $\left|G\right|\leq5 $ then $G$ is abelian. Is it equivalent to this ...
2
votes
1answer
127 views

Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2

Is there any chance to find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)=\mathbb Z_2$? ($\mathbb Z_2$ is the cyclic group of order $2$) Notation $\operatorname{Aut}...
2
votes
2answers
95 views

Finite abelian groups of odd order

I am reading this paper. It is about finite abelian groups of odd order. I need to find maximal subset which doesn't contain 3-term arithmetic progression. I don't understand the need of odd order. I ...
2
votes
3answers
77 views

If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.

If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$. I am stuck at the first step, unable to figure out how to start. I am ...
4
votes
3answers
181 views

Examples of loops which have two-sided inverses.

Are there any neat examples of non-associative loops such that for each element a in the loop there exists $a^{-1}$ so that $a*a^{-1}=1=a^{-1}*a$. Even cooler would be a commutative loop. Also: are ...
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0answers
55 views

Question about finitely generated abelian groups

Why is an Abelian group finitely generated iff $A/mA$ is finite for some $m\gt 1$ and $A$ has a norm function? I know that $mx$ where $x$ is an element of $A$ is equivalent to $0$ in $A/mA$, and I ...
3
votes
1answer
81 views

If $f\in\hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$.

Prove that if $f\in \hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$. I took an element of $\prod_{i=1}^{\infty }\...
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vote
1answer
118 views

Torsion subgroups of finitely generated abelian groups

What information one can get about the torsion subgroups from a short exact sequence of finitely generated abelian groups ?
3
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1answer
96 views

Fibered coproducts in $\mathsf{Ab}$

Aluffi (II.3.9) asks Show that fiber products and coproducts exist in $\mathsf{Ab}$. (Cf. [exercise on fiber products and coproducts in $\mathsf{Set}$]). The equalizers for $\mathsf{Ab}$ and $\...
4
votes
1answer
44 views

Can this be proved purely on base of UMP?

Let $A,B$ be abelian groups and let $P$ serve as a product with projections $p_{A}:P\rightarrow A$ and $p_{B}:P\rightarrow B$. Let $C$ be an abelian group and let $f:C\rightarrow A$ and $g:C\...
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vote
2answers
105 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
1
vote
1answer
145 views

Prove a group is an abelian group

Let $G\subseteq \mathbb N.$ How do I prove that $G$ is an abelian group with respect to the binary operation " * " defined by $\;a*b = a+b+11$ ?
3
votes
3answers
112 views

Are $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? [closed]

Are the groups $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? Why or why not? (Here $\times$ means the direct product or direct ...
4
votes
1answer
69 views

Computation of a homology group of a simple complex $0\rightarrow \mathbb{Z}^l \rightarrow \mathbb{Z}^n \rightarrow \mathbb{Z}^m\rightarrow 0$

Consider the following sequence of abelian groups, where $f\circ g = 0$. $$0\longrightarrow \mathbb{Z}^l \overset{g}{\longrightarrow}\mathbb{Z}^n\overset{f}{\longrightarrow}\mathbb{Z}^m\...