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

learn more… | top users | synonyms

4
votes
1answer
100 views

Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
3
votes
1answer
114 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
0
votes
1answer
225 views

Using the conjugacy class equation [duplicate]

Let $G$ be a group of order $p^2$. Use the class equation to prove that $G$ is abelian. The conjugacy class equation, at least how I remember it, is $$ |G| = |Z(G)|+\sum_{x\in I \backslash Z(g)} \...
3
votes
2answers
178 views

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian.

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian. I KNOW THERE IS A QUESTION OF THE SAME NAME. However, I need more involved assistance. My ...
2
votes
1answer
77 views

Is an abelian group the sum of a torsion group with a free group?

Is an abelian group $G$ always isomorphic to the direct sum of its torsion group $T$ with a free group? Alternatively, is the quotient $G/T$ always free? By definition, $T$ consists of all elements ...
3
votes
1answer
92 views

Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
0
votes
2answers
121 views

Abstract Algebra. Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$.

Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$, called the torsion subgroup of $\mathit{G}$. let $g \in G$ I know that ...
2
votes
1answer
833 views

Classifying abelian groups up to isomorphism

List all abelian groups (up to isomorphism) of the given orders: a) $144$, b) $600$ a) For order $144$, I feel confident with this one so far: $\mathbb{Z}_4 \oplus \mathbb{Z}_{36}$ Elementary ...
1
vote
1answer
552 views

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 ...
5
votes
1answer
226 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 ...
0
votes
2answers
52 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 ...
1
vote
1answer
62 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 ...
9
votes
2answers
220 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
votes
1answer
133 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 ...
5
votes
1answer
124 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
votes
1answer
253 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
votes
1answer
225 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 ...
2
votes
3answers
476 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 ...
1
vote
1answer
117 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 ...
0
votes
3answers
93 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. ...
1
vote
3answers
117 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
votes
1answer
59 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
1
vote
2answers
60 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
votes
1answer
193 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 ...
1
vote
0answers
70 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 $n$ ...
1
vote
0answers
54 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 $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$?
2
votes
2answers
171 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
votes
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
votes
2answers
135 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
159 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
vote
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
vote
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
140 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
70 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
88 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
991 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
votes
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
vote
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
56 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
172 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 ...