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
63 views

Determine the isomorphism class of M/T(M)

Let $M=\Bbb{Z}\oplus\Bbb{Z}\oplus\Bbb{Z}$ and $T: M\rightarrow M$ given by $T(x,y,z)=(4x+2z,2y,2x+10z)$. Show the cokernel $M/T(M)$ is an abelian group of order $72$, and determine its isomorphism ...
3
votes
2answers
67 views

Is this a proof by contradiction?

Below is a proof that any group of order $p^2$ is abelian $(p$ prime of course). Let $Z \left({G}\right)$ be the center of $G$. We know $|Z(G)|>1$. $\color{blue}{\text{Suppose}} \left\vert{Z ...
4
votes
1answer
56 views

$\biggl ( \prod_p G_p \biggr) /\biggl( \bigoplus_p G_p\biggr)$ is divisible

Let $G$ be an abelian group, $p$ a prime, then $G_p$ is the $p$-primary component of $G$, i.e. $$G_p = \lbrace g \in G \ | \exists \ n \in \mathbb{N} \ , p^ng = 0\rbrace$$ I have to prove that ...
4
votes
1answer
288 views

Showing that any group of order 286331153 is abelian

This is the third part of a set of problems, of which I have solved 2. I have shown that if $p$ is prime, the group $Aut(\mathbb Z_p)$ is of order $p-1$. I have shown that $Aut(\mathbb Z_{17})$, ...
2
votes
0answers
24 views

Counting homomorphisms by the order of their images

I am trying to count homomorphisms from $\mathbb Z^r$ to $(\mathbb Z/m)^n$ while keeping track of the order of the image of each map. In other words, for each integer $k$ dividing $m^n$, I want to ...
2
votes
4answers
100 views

Homomorphism from $\mathbb{Z}\oplus \mathbb{Z}_2$ to $\mathbb{Z}$.

What sort of homomorphisms can I have from $\mathbb{Z}\oplus \mathbb{Z}_2$ to $\mathbb{Z}$? What about if I know the homomorphism sends the $\mathbb{Z}$ part to zero. In other words, if I know my ...
4
votes
2answers
260 views

Simple proof of the structure theorems for finite abelian groups

Many proofs of the structure theorems for finite abelian groups first reduce to the problem to $p$-groups, which is fine and is an important technique. However, it seems to me that a simple proof can ...
1
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1answer
63 views

automorphism group of a group of order $p^2$, where $p$ is prime

There is a corollary that states: "If $|P|=p^2$ for some prime $p$, then $P$ is abelian. More precisely, $P$ is isomorphic to either $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_p\times \mathbb{Z}_p$." I know ...
1
vote
1answer
107 views

showing that a group of order 45 is abelian

I'm trying to understand the following proof from Dummit & Foote (pg. 137) which shows why a group of order 45 is abelian. I understand everything but the last two sentences. Why is it that ...
0
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0answers
37 views

Using Smith Normal form to determine isomorphism

For a finitely generated Abelian group (with a relation matrix $R$ given) $A_R = <a1,a2,a3,a4 | R\ast a=0>$, determine the structure of $A_R$. I have already reduced the relation matrix into ...
5
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0answers
40 views

What are the invariant factors of $(\mathbb{Z}/(1000))^\times$?

I'm curious about the invariant factors of $(\mathbb{Z}/(1000))^\times$. I put down $$ (\mathbb{Z}/(1000))^\times\cong(\mathbb{Z}/(8))^\times\oplus(\mathbb{Z}/(125))^\times $$ It's easy to compute by ...
3
votes
1answer
31 views

Orbits under action of a subgroup on the set of conjagtes of a second subgroup

i have the following question: Let $A\leq B\leq G$ be finite groups. Then $G$ acts naturally via conjugation on the set of conjugates $A^G$. It's trivial, that there is only one orbit under this ...
2
votes
3answers
104 views

A sequence of subgroups tending to the trivial subgroup

Do you have an example of an abelian group $G$ with a sequence of mutually distinct nontrivial subgroups $(A_n)$ such that $$\dots \le A_n\le\dots \le A_2\le A_1\le A_0=G$$ and ...
5
votes
1answer
75 views

The splitting lemma and uniqueness

For the sake of concreteness, let's restrict discussion to the category of abelian groups. Throughout, $$ 0 \to A \overset{q}{\to} B \overset{r}{\to}C \to 0$$ is a short exact sequence. One part of ...
1
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2answers
31 views

Question about order of product of elements in a group

Let G be a finite abelian group. Prove that the product of all elements in G has order 2. I think i am supposed to use lagrange's theorem but how?
0
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1answer
37 views

Cyclic and abelian groups

Just looking for the criteria which I would use to say if these groups are cyclic. Like a short proof? for (i), (ii), (iii), (iv) (v) Thank you.
3
votes
1answer
78 views

Automorphisms of abelian groups and Choice

The latest question to be asked at the Group Pub Forum is a classic: can every group be realised as the automorphism group of a group? The answer is no, and the canonical answer is the infinite cyclic ...
0
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0answers
96 views

Finding new generators for a finite Abelian Group

Let $A_R:= \langle a_1, a_2, a_3, a_4 | R \circ \underline{a} = 0\rangle $ be the finitely generated abelian group, determined by the relation-matrix $R :=$ $$ \begin{bmatrix} -6 & 111 & ...
0
votes
1answer
43 views

If $G$ has more than one nontrivial elements with order 2, how to show that $\prod^n_{x=1}a_x=1$? [duplicate]

Let $G$ be an abelian group of order $n$, and $a_1,a_2,...a_n$ its elements. If $G$ has more than one nontrivial elements with order $2$, how to show that $\prod^n_{x=1}a_x=1$?
10
votes
1answer
128 views

Prove $G$ is abelian if $f(f(x)) = x$?

Let $G$ be a finite group and $f$ an automorphism such that $f(f(x)) = x$, and $f(x) = x$ if and only if $x=e$. Prove that $G$ is abelian and $f(x) = x^{-1}$. My attempt: ...
2
votes
1answer
48 views

How to find group homomorphisms from one group to another

I am trying to figure out all the homomorphisms from $\mathbb{Z}_2\times\mathbb{Z}_2$ to $\mathbb{Z}_2$. Is there a good process for doing such a think? I am getting lost...
5
votes
2answers
131 views

A non-abelian group such that $G/z(G)$ is abelian.

I'm looking for an example of a non-abelian group $G$ such that $G/z(G)$ is abelian, where $z(G)$ is the center of the $G$. In other words, I'm looking for a non-abelian group where $z(G)$ contains ...
3
votes
1answer
61 views

Subgroups of order $p^2$ present in a abelian group

How many subgroups of order $p^2$ does the abelian group $\mathbb{Z_{p^3}} \times \mathbb{Z_{p^2}}$have ?
4
votes
1answer
190 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
3
votes
1answer
57 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
0
votes
2answers
39 views

Subgroups which are not subspaces

Let $p$ be a prime number. Is every subgroup of the abelian group $\Bbb Z_p^2$ a subspace of it as a vector space over $\Bbb Z_p$? Can it be generalized to all finite fields?
6
votes
2answers
106 views

Rank of $(G/H)/(G/H)_t$ where $G$ is finitely generated abelian and $H$ is a subgroup.

Let $G$ be a finitely generated abelian group and $H$ be a subgroup. Let subscript $t$ denote the torsion subgroup. If $G/G_t$ is free of rank $n$ and $H/H_t$ is free of rank $m$, it is easy to embed ...
2
votes
2answers
162 views

$\langle x \rangle$ is a direct summand of a finite abelian group where $x$ is maximal order [duplicate]

Let $x$ be an element of a finite abelian group $G$ where $x$ has maximal order. Then I want to show that $\langle x\rangle$ is a direct summand of $G$. Note that I do not want to use finite abelian ...
0
votes
2answers
114 views

$\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
2
votes
1answer
77 views

Let $H$ be a normal subgroup of $G$. If $H$ are $G/H$ are Abelian, should $G$ be abelian?

(a) Let $H$ be a normal subgroup of $G$. If $H$ are $G/H$ are Abelian, should $G$ be abelian? Attempt: : There's a counterexample to this claim, $G=D_3$ which is non abelian. But, what could be ...
0
votes
1answer
36 views

Why is $U(12) = U_{4} (12) ~ U_3(12)?$

Why is $U(12) = U_{4} (2)~ U_3(12)$ Attempt: Any subgroup $U_k(n) = \{x \in U(n)~~|~~x \mod k=1 , k ~|~n \}$ Hence : if $U(12) =\{1,5,7,11\}$ then : $U_4(12) =\{1,5\}$ and $U_3(12)=\{1,7\}$ We see ...
1
vote
1answer
50 views

Uniqueness FTOFAG

How do you prove uniqueness for the fundamental theorem of finite abelian groups? The book I'm using has this not very well written proof that I can't follow. So following this proof, I multiply by ...
0
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0answers
36 views

The Basis Theorem for Finte Abelian Groups

I am using Pinter's Abstract algebra book to prove the basis theorem for finite abelian groups (Every finite abelian group is a direct product of cyclic groups of prime power order.) $G$ is an abelian ...
0
votes
1answer
22 views

Abelian groups order help

Let $p$ be a prime number. Find the number of abelian groups of order $p^n$, up to isomorphism when n=2,3, and 5. I know the answer when $n=2$ and 3. And my professor said that there are 7 abelian ...
1
vote
1answer
57 views

direct limit of abelian groups

Let $I$ be a directed set and let $(A_i)_{i \in I}$ be a collection of abelian groups. Let $A = \varinjlim A_i$ be its direct limit. Suppose its maps are $\rho_{ij} : A_i \to A_j$ for $i \leq j$. I ...
2
votes
1answer
98 views

Subgroups of a finite elementary abelian group.

I am looking for a method to calculate number all subgroups of a finite elementary abelian $p$-group. Suppose $G$ be an elementary abelian $p$-group of order $p^n$. A proper subgroup $H$ of $G$ is ...
0
votes
0answers
31 views

Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k $ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
0
votes
1answer
22 views

Evidence about the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$

Be $p$ an odd prime number. Show that the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$ has a unique element of order $2$, namely $\overline{p-1}$, and show that ...
1
vote
1answer
26 views

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$, $|G|=n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$.

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$ an abelian group of order $n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$. I have serious difficulties with ...
0
votes
1answer
193 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
5
votes
2answers
136 views

Equivalence relation to make a group commutative

A while ago I was wondering if there is a "natural" way to make a commutative group out of an arbitrary one. I played with the idea a bit and here is what I came up with. Define a binary relation ...
3
votes
2answers
132 views

If the intersection of a normal subgroup and the derived group is {e}, show that N is a subset of Z(G).

I think my reasoning is wrong, but if the intersection only contains the identity, doesn't that imply that the only commutator in N is {e}, so doesn't that mean N is automatically commutative? Why was ...
2
votes
2answers
75 views

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
2
votes
1answer
84 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
1
vote
1answer
58 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
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votes
1answer
54 views

conjugacy classes and order of group

Suppost that $k_G(A)$ denotes the number of conjugacy classes of $G$ that intersects $A$ non-trivially ($A$ is an arbitrary subset of $G$) and $M=G^{'}Z(G)$. Also suppose that $G$ is non-solvable, ...
0
votes
0answers
33 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
2
votes
1answer
65 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
1
vote
0answers
108 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
1
vote
4answers
17 views

Subset of elements of a given order in a group

I'm almost certain this is true (I have even given a proof) but I keep getting this strange feeling that something is not quite right so I will ask... Let $G$ be an abelian group and let $r$ be a ...