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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1answer
147 views

If every proper quotient is finite, then $G\cong\mathbb Z$

Here is my problem: Let $G$ is an infinite abelian group. Prove that if every proper quotient is finite, then $G\cong\mathbb Z$. And here is my incompleted approach: I know that the quotient ...
2
votes
2answers
182 views

The ring of formal power series

Is there a simple proof/clarification of this statement? The set of all formal power series in X with coefficients in a commutative ring R form another ring that is written R[[X]], and called the ...
22
votes
1answer
707 views

Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How ...
0
votes
1answer
245 views

Pure Subgroups of $\mathbb Z\times\mathbb Z$

I am thinking of the following problem $^*$: Given an example in which the subgroups generated by two pure subgroups ia not pure. (Hint: Look within a free abelian group of rank $2$.). So, as ...
1
vote
2answers
81 views

Bases for $\Bbb Z^n$ containing a given vector

I am trying to prove the following theorem on finitely generated free abelian groups (which thus for simplicity may be assumed to be $\Bbb Z^n$): Let $\alpha \in \Bbb Z^n$ be such that for all $k ...
4
votes
1answer
123 views

What do you call groups where you can apply the group operation a fractional number of times?

I have a group $G$, and for all $g \in G$ and $a,b \in \mathbb{Z}$ it makes sense to talk about the element $g^{a \over b} \in G$. To get some intuition, I've been thinking about what it would mean ...
0
votes
3answers
759 views

G is an abelian group. Prove $G^{(n)}$ is a subgroup of G

Let G be an abelian group. Prove that $$G^{(n)} = \{g \in G | g^n = 1_G \}$$ is a subgroup of G. How do I go about doing this? I understand that $G^{(n)}$ is basically the set of all elements ...
1
vote
2answers
61 views

Commutative Groups and Quotients

Let (A,+) and (B,+) be commutative groups and suppose that A is isomorphic to B. Prove that $A/dA$ is isomorphic to $B/dB$, where $d \in \mathbb{N}$. Any thoughts on this one?
4
votes
3answers
312 views

Frattini subgroup of additive group of rational numbers

Show that Frattini subgroup of additive group of rational numbers $(\mathbb{Q},+)$ is itself, or $$\Phi(\mathbb{Q})=\mathbb{Q}$$ PS. My strategy is prove that group $(\mathbb{Q},+)$ hasn't maximal ...
3
votes
2answers
328 views

Direct limit of $\mathbb{Z}$-homomorphisms

What is the direct limit of the following sequence of $\mathbb{Z}$-homomorphisms (as groups)? $$ \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{3} \mathbb{Z}\xrightarrow{5} ...
1
vote
1answer
62 views

$G/S$ is torsion-free?

There is a well-known theorem that: If $S\le G$ and $\frac{G}{S}$ is torsion-free, so $S$ is pure in $G$. Please hint me about the reverse. If $S$ is pure in $G$, then will $\frac{G}{S}$ is ...
6
votes
0answers
57 views

Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
2
votes
1answer
306 views

p-group: cyclic $n \leq 1$ | abelian $n \leq 2$

$p$ prime number, $n$ a non-negative integer, $G$ group. (a) $\forall G$ with $|G| = p^{n}$ cyclic $\Leftrightarrow$ $n \leq 1$. (b) $\forall G$ with $|G| = p^{n}$ abelian $\Leftrightarrow$ $n \leq ...
1
vote
2answers
87 views

inverse of even number of elements in a group

if an abelian group with |G|=n where n is odd. if i take out the identity i'm left with even # of distinct elements. can this mean that each element has an inverse which is not itself?? not a homework ...
0
votes
1answer
158 views

Group homomorphisms between two abelian groups with different kernel

Does there exist two abelian groups $A,B$ with an epimorphism $f: A\to B$, and two other abelian groups $A', B'$ along with an epimorphism $g: A'\to B'$ such that $A\cong A'$, $B\cong B'$ and $ker\,f ...
6
votes
1answer
145 views

$G$ is isomorphic to a subgroup of $H$ and vice versa

Let $G$ and $H$ are two divisible groups that each of which is is isomorphic to a subgroup of the other, then $G\cong H$. What I've done is to use the injective property for both groups: ...
3
votes
0answers
272 views

Direct sum of Prüfer groups and $\mathbb Q/\mathbb Z$

It can be easily shown that, the Prüfer $p$-group $\mathbb Z(p^\infty)$ is isomorphic to multiplicative group $$R_p=\{e^{2\pi ik/p^n}|k\in\mathbb Z,n\geq0\}$$ Now I want to prove that: ...
1
vote
2answers
131 views

$G$ is a reduced Group

Here is the problem: Let $G=\langle x_0,x_1,x_2,\ldots\ |px_0=0,x_0=p^nx_n, \text{all } n\geq1\rangle$. Prove that $G/\langle x_0\rangle$ is a direct sum of cyclic groups and is reduced. The ...
4
votes
2answers
187 views

Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?

I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
1
vote
0answers
72 views

If $G=\prod_{p\in P}\mathbb Z_p$ then $\frac{G}{tG}$ is divisible.

I want to show that: If $G=\prod_{p\in P}\mathbb Z_p$, wherein $P$ is the set of all primes, then $\frac{G}{tG}$ is divisible. I know that $tG$ is not a direct summand and if $x\in G$ wants to ...
1
vote
1answer
84 views

$F/H$ has an element of infinite order?

Here is a problem: If $F$ is a free abelian group of rank $n$ and $H$ is a subgroup of rank $k<n$, then $F/H$ has an element of infinite order. What I did: I assume $F=\langle ...
2
votes
1answer
100 views

$\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$

I want to prove that: If $F$ is a free abelian group of rank $n$, then $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$ with ...
2
votes
1answer
86 views

Verifing some properties about $G$

I have the following problem$^*$: Prove that the group $G$ having generators and relations respectively $$X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$$ is an infinite ...
1
vote
1answer
58 views

$G$ is torsion-free group then $G/\langle X\rangle$ is torsion

Honestly, I have been thinking on this problem for hours but couldn't find a way: Let $G$ is torsion-free group and $X$ is a maximal independent subset, then $G/\langle X\rangle$ is torsion. I ...
1
vote
1answer
110 views

Divisible abelian $q$-group of finite rank

What does "finite rank" mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
1
vote
1answer
69 views

Exact sequence of abelian groups, property transfers

We had the statement that with an exact sequence of multiplicatively written abelian groups $U \mapsto V \mapsto W \mapsto X \mapsto Y$ and in $U$, $V$, $X$, $Y$ every group element has a unique ...
2
votes
2answers
142 views

Prove $H \times G$ is commutative iff $H, G$ are commutative

This is another proof question I am asking about, can someone give me tips on how to answer these questions? My question says: "Let $H,G$ be arbitrary groups. Prove that $H \times G$ is commutative ...
1
vote
2answers
861 views

determining number of subgroups

If $G$ is an abelian group of order 72, do we know how many subgroups of order 8 it has? Just because it's a divisor doesn't mean that there is a subgroup of that size. But I'm wrong. Why?
1
vote
1answer
259 views

Groups of the same order that are nonisomorphic

I'm reading A First Course in Abstract Algebra by Fraleigh and I've reached a point where I feel like I'm supposed to have understood something more from the chapter than what is actually stated. I've ...
4
votes
1answer
159 views

Smallest pure subgroup containing a fixed subgroup

I will ask a slightly more precise question then in the title. Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
3answers
288 views

Splitting exact sequences of finite abelian groups

I would like to find a condition for an exact sequence of abelian groups $$ 0\to H\to G\to K\to 0 $$ to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
4
votes
1answer
456 views

An indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable

I'd like to find an indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable, and I'm running out of ideas: The only indecomposable $\mathbf{Z}$-modules I know are $\mathbf{Z}$, ...
3
votes
3answers
205 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
0
votes
2answers
76 views

Elementary Question about Torsion Subgroups

Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$. Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime. Denote by $G[a]$ resp. $G[b]$ the $a$-resp. ...
4
votes
1answer
237 views

When abelian group is divisible?

By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$. Let $A$ be abelian group with no proper subgroups of finite index. How can I prove ...
3
votes
1answer
162 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
5
votes
1answer
367 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...
0
votes
0answers
61 views

Abelian group problem with three consecutive intergers [duplicate]

Possible Duplicate: Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian If $G$ is a group such that $ (a \circ b)^i = a^i \circ b^i $ for ...
5
votes
2answers
230 views

Subgroups of $\Bbb{R}^n$ that are closed and discrete

I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$. Base ...
6
votes
6answers
5k views

Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
8
votes
3answers
192 views

Tensor-commutative abelian groups

Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$. The first question is somewhat vague: Question 1. ...
2
votes
1answer
64 views

The $p^k$-rank of a subgroup is no greater than the $p^k$-rank of the group.

Recently I was given a handout containing (roughly) the following text: Let $A$ be a finite abelian group, and $p^k$ a prime power. The $p^k$-rank of $A$ is defined to be ...
4
votes
1answer
257 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
3
votes
1answer
283 views

exponent of an abelian group

Let $p$ be a prime. Let $H_{i}, i=1,...,n$ be normal subgroups of a finite group $G$. I want to prove the following: If $G/H_{i}$, $i=1,...,n$ are abelian groups of exponent dividing $p−1$, then $G/N$ ...
5
votes
1answer
300 views

Why do characters on a subgroup extend to the whole group?

As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally ...
1
vote
0answers
69 views

Pontrjagin's Lemma and an application

I would appreciate any kind of help on the following issue: On page 114 of Rotman's "Homological Algebra", exercise 3.4 reads: 1) (Pontrjagin) If an abelian group $A$ is countable, torsion-free ...
3
votes
1answer
148 views

CEP for Abelian groups and lattices

An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta \in \operatorname{Con} B$ there is a $\phi \in \operatorname{Con} A$ such that $\theta = \phi \cap (B\times ...
-3
votes
2answers
761 views

Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
3
votes
1answer
489 views

Torsion-free quotient group of an abelian group

Let $G$ be an abelian group, and let $H\leq G$. Prove that if $G/H$ is torsion free, then $H$ contains the torsion group of $G$. Proof: Let $x\neq1$ be an element in the torsion group. Thus there ...
3
votes
3answers
272 views

Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element

Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$: $$\begin{aligned} a \odot b=a+b+3\end{aligned}$$ Check if $(\mathbb Z_7, \odot)$ is a group and ...