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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Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
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1answer
21 views

Proof about existence of subgroups in finite abelian group

I am trying to understand the proof of the following theorem. Theorem: If $G$ is a finite abelian group and $d$ is a divisor of $|G|$, then $G$ contains a subgroup of order $d$. Proof: Let ...
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622 views

What does it mean for a group to be Abelian?

I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
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173 views

Normal Abelian Subgroup does not imply Abelian Quotient Group

I'm a bit confused and just need some clarification about what I am missing in this: I have $S_4$ with normal subgroup $N=\lbrace(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\rbrace$. I know that $N$ is ...
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42 views

the torsion subgroup of E(Q) (eliptic curves)

if $E$ is an elliptic curve over $Q$, then why $E(Q)_{\rm tor}$ is group and finite set ?
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$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
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33 views

homomorphisms between amenable (discrete) groups

Let $\theta \,: G \to H$ be a group homomorphism between amenable groups (s.t. $\theta(G)$ is a normal subgroup of $H$, if needed). Is it possible to define amenable means $m_G$ on $L^\infty(G, ...
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74 views

Is my solution correct? Finite abelian groups are CLT groups.

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each ...
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1answer
43 views

Nilpotent group with torsion divisible abelian quotient

Just want to make sure this is true: If $G$ is a nilpotent group such that $G/[G,G]$ is a torsion divisible abelian group (like $\mathbb{Q}/\mathbb{Z}_{(p)}$), then $G$ is abelian. I get that ...
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38 views

A proposition on exact sequence of inverse limit (Lang, Algebra, p. 165)

I am trying to understand this proof. My only question is that what are the vertical maps here?
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64 views

Proof of closure and identity element in Abelian Group

For real numbers $x > 1$, which forms the set $G$, it is given that the operation on $a,b$, being $a\ast b$, results in $ab - a - b + 2$ (where $ab$ is the ordinary multiplication of $a$ and $b$). ...
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60 views

A doubt in Atiyah-Macdonald's “Introduction to Commutative Algebra”

"Introduction to Commutative Algebra" by Atiyah-Macdonald says the following: Let $G_n$ be the subgroup containing elements of order $p^n$ in the group $\Bbb{Q/Z}$ for all $n\in\Bbb{N}$. Here $p$ ...
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32 views

Show that $ℤ^{m}$ is a subgroup (and a free abelian group) of $ℤ^{n}$ for all $m≤n$

My question is: Show that $ℤ^{m}$ is a subgroup (and a free abelian group) of $ℤ^{n}$ for all $0≤m≤n$.
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18 views

list of conjugacy classes in elementary abelian p-group

Let G be an elementary abelian p-group, how can I get a complete list of conjugacy classes in G? A general structure of the conjugacy classes will do. Thank you in advance. Magero Fidelius
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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 ...
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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 ...
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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 ...
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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 $Aut(\mathbb Z_{17})$, ...
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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 ...
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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 ...
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220 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 ...
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61 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 ...
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66 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 ...
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28 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 ...
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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 ...
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30 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 ...
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103 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 ...
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1answer
73 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 ...
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30 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?
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35 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.
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76 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 ...
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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 & ...
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38 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 $\prod^n_{x=1}a_x=1$?
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127 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: ...
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1answer
47 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...
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121 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 ...
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1answer
58 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 ?
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182 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 ...
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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. ...
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38 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?
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105 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 ...
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153 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 ...
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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 ...
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1answer
76 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 ...
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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 ...
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48 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 ...
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34 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 ...
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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 ...
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51 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 ...
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97 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 ...