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 any subgroup of a finitely generated abelian group is finitely generated?

I am working through Rotman 2.89 and I can't seem to solve this one. Note: Please do not link me to the related questions such as Proving that a subgroup of a finitely generated abelian group is ...
3
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1answer
123 views

Finite abelian p-group and an element of maximal order

I'm studying for an exam and I'm having trouble understanding the proof given for the following statement: Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there ...
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2answers
44 views

Basic proof of statement in abstract algebra?

http://www.proofwiki.org/wiki/Abelian_Quotient_Group The third step (in both proofs) is something I am having trouble seeing. The theorem itself is not difficult to prove, but it is much cleaner this ...
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1answer
37 views

Prove that the following are equivalen for abelian group

Let $(G, *)$ be a group. Prove that the following are equivalent: a. $G$ is abelian. b. $aba'b' = e$ for all $a,b \in G$. c. $(ab)^{2}$ = $a^2b^2$ for all $a, b \in G$.
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1answer
52 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
7
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1answer
116 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
3
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1answer
111 views

Is there a name for the generalization of the concept “Abelian group” where the axiom $-x+x = 0$ is weakened to the following?

Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list? $−0=0$ $−(x+y)=−x+−y$ $−(−x)=x$ $x+(-x)+x = x$ In multiplicative ...
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1answer
88 views

Splitting of exact sequence of groups when middle group has split subgroup.

I am trying to show that a short exact sequence of abelian groups splits. I have a short exact sequence, $$0\rightarrow \mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}_2 \rightarrow 0$$ and I know ...
0
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1answer
205 views

Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order 2 or that $G$ has more than one element of order 2. ...
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1answer
145 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
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3answers
120 views

Commutator Subgroup is Normal Subgroup of Kernel of Homomorphism

Please help to understand this problem. Let $G$ be a group, $H$ an abelian group, $\phi : G \rightarrow H$ a homomorphism. Show that $C(G) \lhd \mathrm{Ker}(\phi)$ I must be misunderstanding ...
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1answer
76 views

Prove that a group is abelian if every element commutes with exactly K other elements

Let $(G,*)$ be a finite group with the property that every element, aside the neutral element $e$, commutes with exactly K other elements. Alternatively speaking, the centralizer of every element is a ...
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2answers
57 views

“Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$”

I have a question that says this: Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups. ...
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3answers
77 views

An abelian group is finite $\iff$ the kernel of a surjective homomorphism has rank $n$

I'm doing a course on lineare algebra and I have to show the following: let $H$ be a finitely generated abelian group and $g: \mathbb{Z}^n \to H$ a surjective homomorphism. I want to show that ...
0
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1answer
53 views

If a group has only one commutator, why does that mean it is abelian?

I understand that if $aba^{-1}b^{-1} = e$ then $ab$ is commutative, but I don't see how having multiple commutators will prevent the group from being abelian
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2answers
36 views

Is torsion subgroup of elliptic curve birationally invariant?

It's probably a very basic question: Having two birationally equivalent elliptic curves over $\mathbb{Q}$ - is the torsion subgroup unchanged under the birational equivalence?
3
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1answer
118 views

an order of automorphism group of finite abelian group

This is problem of Rotman's Exercise 7.9(i). If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order. How can I approach to this problem? Could you suggest ...
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0answers
61 views

Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
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0answers
47 views

Find which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$

Which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$?
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2answers
93 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 ...
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0answers
37 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
63 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 ...
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2answers
66 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 ...
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1answer
57 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 ...
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1answer
95 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 ...
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0answers
123 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
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3answers
99 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
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1answer
50 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) ...
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1answer
65 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 $n$, ...
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1answer
82 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. $gh = hg$ ...
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2answers
94 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 ...
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0answers
23 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
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0answers
16 views

Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G $is elementary and ...
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1answer
29 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...
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2answers
44 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 ...
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1answer
65 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
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1answer
42 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 ...
3
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3answers
54 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
80 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 $a_1,...,a_s \in G$. ...
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2answers
152 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 implies $\text{Aut }G$ is cyclic?
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18 views

Quaternions linear

Are the Matrices $I_{4n}$ and $J_{4n}$ defined below quaterionic linear? $I_{4n}$ and $J_{4n}$ are defined where $\mathbb H$->($g2_n(f_n)) R^{4n}$->$R_{J_{4n}}$ where I and J are interchangeable.
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10 views

Quatornianic Matrix are Complex, Proposition

In my matrix group textbook, I was looking at a proposition that said For all $\lambda \in \mathbb R$ and $A, B \in M_n(\mathbb H)$, $\Psi_n(\lambda \bullet A) = \lambda \bullet \Psi_n(A)$ Is this ...
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1answer
24 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.
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1answer
75 views

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? [closed]

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? I tried to find a generator. and i didn't found one. But how can i prove that this group is not cyclic?
0
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0answers
61 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 ...
1
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1answer
78 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
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1answer
57 views

$HK$ is a subgroup of $G$ when $G$ is finite abelian and $H$ and $K$ are subgroups of $G$ [closed]

Let $G$ be a finite group and $H, K$ subgroups of $G$. Let $HK=\{hk\mid h\in H, k \in K\}$. Show that if $G$ is abelian then $HK$ is a subgroup of $G$.
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2answers
34 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 ...
4
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1answer
107 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 ...
0
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3answers
118 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 ...