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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2answers
133 views

Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
6
votes
4answers
588 views

Is $G/pG$ is a $p$-group?

Jack is trying to prove: Let $G$ be an abelian group, and $n\in\Bbb Z$. Denote $nG = \{ng | g\in G\}$. (1) Show that $nG$ is a subgroup in $G$. (2) Show that if $G$ is a finitely ...
0
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2answers
396 views

The group homomorphism from Z to Q*

Let F be non trivial group homomorphism F: Z -> Q*. Want to prove that either Ker(F)={0} or Ker(F)= 2Z. Okay here is what i did; since i know that if there is homomorphism between two groups then ...
0
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1answer
147 views

Homomorphisms between sets of homomorphisms

Let A,B be finite, commutative groups. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$, the set of homomorphisms from $A$ to $\mathbb{Q}/\mathbb{Z}$. $A^{*}$ is abelian itself (take this for granted). Let ...
2
votes
1answer
134 views

$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$

Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
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1answer
110 views

Prove abelian non-cyclic p-group of order p^m (m>2) has a non-cyclic proper subgroup without structure theorem

It should have a copy of $\Bbb Z/p\Bbb Z\times \Bbb Z/p\Bbb Z$ but my brother's class has not learned it yet and this is for a homework problem... Any help?
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2answers
150 views

Existence of certain homomorphisms between $\mathbb{Q}$ and $\mathbb{Q}^{\times}$

Letting $\mathbb{Q}^{\times}$ be the group of non-zero rationals under multiplication, and $\mathbb{Q}$ the additive group of rationals, I am seeking to find the following types of homomorphisms (if, ...
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2answers
106 views

Homomorphism $\mathbb{Z}^2 \to \mathbb{Z}^2$ and g.c.d.

Let $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ be a group homomorphism and suppose that $f(a,b) = (c,d)$. Prove: $\gcd(a,b) \mid \gcd(c,d)$. Prove: $\gcd(a,b) = \gcd(c,d)$ if $f$ is an ...
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votes
3answers
154 views

Subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$

what are the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_{12}$ of order $6$? I know that there are three such subgroups, and two subgroups are clear to me, namely the subgroup isomorphic to ...
2
votes
2answers
394 views

What is the largest order of element in an abelian group of order $5!$?

What is the largest order of element in an abelian group of order $5!$ ? I don't know how to deal with problem. Can anyone give a solution or useful resources?
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3answers
274 views

$G$ group, $H \trianglelefteq G$, $\vert H \vert$ prime, then $H \leq Z(G)$

Let $G$ be a finite group. Let $H \trianglelefteq G$, with $\vert H \vert = p$, a prime, where $p$ is the smallest prime dividing $\vert G \vert$. Prove that $H \leq Z(G)$. (Hint: If $a \in H$, by ...
2
votes
1answer
756 views

In an abelian Group in which there are 2 subgroups of order m and n there is a subgroup of order mcm(m,n), how to prove?

This is an Herstein's exercise If an abelian group has two subgroups of order m and n, prove that it also has a subgroup of order lcm(m,n). I've solved the exercise right before this, which is ...
3
votes
1answer
144 views

The cancellation property for finite abelian groups

I need some hints to prove that: Let $A,B,C$ are finite abelian groups such that $A\oplus B\cong A\oplus C$. Prove that $B\cong C$. I know that every finite abelian group can be written as a ...
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1answer
220 views

Abelian Groups of order 2000

Classify, up to isomorphism, all abelian groups of order 2,000, giving the standard form of each group in your list. (The standard form is also called the invariant factor decomposition.)
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3answers
119 views

Suppose $G$ Abelian and $f:G\rightarrow \Bbb Z$ is surjective with kernel K. Show $G \cong H + K$ where $H \cong \Bbb Z$

Suppose $G$ abelian and $f:G\rightarrow \mathbb Z$ is surjective with kernel $K$. Show that $G$ has a subgroup $H$ such that $H \cong \mathbb{Z}$ Show that $G \cong H\bigoplus K$ To get ...
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1answer
445 views

Group of homomorphisms

Let $A$ be a finite, Abelian, additive group. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$ denote the group of homomorphisms $f$ from $A$ to $\mathbb{Q}/\mathbb{Z}$. Take for granted that $A^{*}$ is an ...
0
votes
1answer
90 views

Number of subgroups of order p

Let p be a prime number and consider $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$. How many subgroups of order p does it have? Given any two subgroups $B_1, B_2 $ of ...
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vote
1answer
79 views

Seeking a proof of: If any two Abelian groups of order $d$ are isomorphic, then $d$ is squarefree.

Suppose that $d \in \mathbb{N}$ satisfies the property that given any two Abelian groups $A_1, A_2$ of order $d$, $A_1 \cong A_2$. Prove that given any prime $p$, $p^2 \nmid d$. What can one say for ...
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votes
1answer
563 views

Non-trivial homomorphism between multiplicative group of rationals and integers

Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
0
votes
1answer
77 views

showing G is abelian

If $G$ group of order $52$ includes a normal group of order $4$ then $G$ is abelian. I did like this $$ |G|=52=2^2\cdot 13 $$ let $H$ be normal group of order $4$. $n_{13}=1$ thus $G$ has a $K$ ...
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votes
2answers
221 views

How to show that these groups are not isomorphic?

Let $G$ is an abelian group and $tG$ is its torsion subgroup. If $p$ is a prime, how to show that: $$t\bigg(\prod_{n=1}^{\infty}\mathbb Z_{p^n}\bigg)\ncong\sum_{n=1}^{\infty}\mathbb Z_{p^n}$$ I ...
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
183 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
715 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 ...
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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 ...
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3answers
773 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 ...
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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
314 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
330 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} ...
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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
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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
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1answer
318 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 ...
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2answers
94 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
161 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
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0answers
274 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: ...
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2answers
135 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 ...
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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} * ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
886 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?
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1answer
260 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 ...