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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3
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3answers
111 views

Are $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? [closed]

Are the groups $\Bbb Z_{8} \times \Bbb Z_{10} \times \Bbb Z_{24}$ and $\Bbb Z_{4} \times \Bbb Z_{12} \times \Bbb Z_{40}$ isomorphic? Why or why not? (Here $\times$ means the direct product or direct ...
4
votes
1answer
69 views

Computation of a homology group of a simple complex $0\rightarrow \mathbb{Z}^l \rightarrow \mathbb{Z}^n \rightarrow \mathbb{Z}^m\rightarrow 0$

Consider the following sequence of abelian groups, where $f\circ g = 0$. $$0\longrightarrow \mathbb{Z}^l ...
6
votes
0answers
188 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
-2
votes
1answer
392 views

Prove that is an abelian group

Given the binary operation in $a \circ b \in \mathbb{R}$ defined by $a \circ b = a + b + \pi$, prove that with this operation in $\mathbb{R}$, it's an abelian group. I know that: Given a group $G$ ...
4
votes
2answers
111 views

Commutativity of a finite group

In a finite group a representative can be chosen from each conjugacy class such that they all commutate. Prove that the group is commutative. Does this still hold true if the group is infinite?
4
votes
1answer
258 views

Isomorphic finite abelian groups [duplicate]

Let $G$ and $H$ be finite abelian groups. Show that if for any natural number $n$ the groups $G$ and $H$ have the same number of elements of order $n$, then $G$ and $H$ are isomorphic. I know, ...
1
vote
1answer
88 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
0
votes
3answers
242 views

show that rational numbers with the multipiciation are not abelian finitely generated group

we need to show that $( Q \ast , \bullet )$ is not abelian finitely generated group for all finite subset $S \subseteq Q \ast $ $ Q\ast=Q \setminus \big\{0\big\} $
2
votes
2answers
120 views

Let $A$ be a finitely generated abelian group. Show that $\operatorname{Hom}(A,Z)$ is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
1
vote
1answer
100 views

Let $A$ be an abelian group. Show that $\mathrm{Hom}(\mathbb Z, A)$ is isomorphic to $A$.

Let $A$ be an abelian group. Show that $\mathrm{Hom}(\mathbb Z, A)$ is isomorphic to $A$. My problem is figuring out how to define Φ and using it show the homomorphism between ...
2
votes
0answers
113 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
4
votes
2answers
501 views

Tensor product of a finitely generated abelian group and the field of rational numbers

Let $G$ be a a finitely generated abelian group. Then $G\otimes_\mathbb{Z} \mathbb{Q} = 0$ if and only if $G$ is a finite group. The "if" part is easy. The "only if" part can be proved using the ...
5
votes
3answers
358 views

An Example of Abelian Group with exactly one maximal subgroup.

Let $G$ be a finitely generated group and $G$ has exactly one maximal subgroup. Then I can conclude that $G\cong\mathbb Z_{p^k}$. Now, I am looking for an example of infinite abelian group $G$ such ...
6
votes
1answer
226 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
3
votes
1answer
153 views

Group with $a=a^{-1}$ for all $a\in G$ is abelian

Let $G$ be a group, and suppose that $a= {a^{-1}}$ for every $a\in G$. Prove that $G$ is Abelian. I know that I need to prove that ${a^{-1}}$ = a and use right multiplication. I also know that I ...
2
votes
1answer
168 views

What motivates the study of Abelian groups?

Monoids arise naturally as endomorphism monoids, and groups arise naturally as automorphism groups. These are among the primary motivators for their study, in my opinion. What are the (main) ...
-1
votes
2answers
156 views

The number of elements of a finite group which is a quotient of a finitely generated free abelian group

Let $G$ be a finitely generated free abelian group. Let $\omega_1,\cdots,\omega_n$ be its basis. Let $\alpha_1,\cdots,\alpha_m$ be a finite sequence of elements of $G$. Suppose $\alpha_i = \sum_j ...
5
votes
6answers
2k views

Prove that a finite abelian group is simple if and only if its order is prime. [duplicate]

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...
1
vote
3answers
93 views

Finding a non-Abelian group

I am stuck at another problem in my homework. Find a non-Abelian group of size 48 such that the order of its elements are either 1, 2, 3 or 6. I need some hints/tips to start on this problem as ...
0
votes
3answers
80 views

Finding an Abelian subgroup

I am working on my homework and am currently stuck at the following question: Let $\mathcal S_{10}$ be the group of all permutations of the set of elements {1, 2, ..., 10}. Find an Abelian ...
1
vote
2answers
129 views

multiplicative order in field

Let $\alpha$ be primitive element of GF(7). Then order of $\alpha$ is 6, i.e. $o(\alpha)=6$. Now we know that $\alpha^4$ is not equal to 1, and that $o(\alpha^4)$ = $\frac{6}{gcd(6,4)}$. This also ...
1
vote
3answers
345 views

Find all possible abelian groups of order $120$.

Find all possible abelian groups of order $120$. If someone could walk me through how to do this, that would be great.
1
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2answers
67 views

Properties of Order of a Group

Having real trouble getting started on this question, even though it doesn't seem hard: Let $g,h \in G$ where $G$ is an Abelian group. Then assume that $ord(g), ord(h)$ are finite with ...
2
votes
3answers
119 views

Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite?

Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite? I suspect it does not. However since we could take a subgroup of all $p$ sets $\{\frac{1}{p} + \Bbb Z\}$ if we consider $p$ to be ...
1
vote
2answers
312 views

How do we prove this fact about cyclic groups?

Prove that an Abelian group of order 33 is cyclic. Can we take an element a of order 3 and an element b of order 11 and say, |ab|=33?
0
votes
2answers
95 views

meaning of the symbol $Z_n^*$ in discrete mathematics

I was reading discrete Mathematics, and i found a symbol $$Z_n^*.$$ I don't know what it means. The text says that the "image" with the multiplication operator is an abelian group. can any one ...
1
vote
1answer
62 views

Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?

Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?
0
votes
2answers
44 views

Isomorphism of rings under different operations

i am new to this site. i have been reading through its posts and question and they are really amazing. however, i found a link to a question asked about one year ago and a question i don't know how to ...
3
votes
2answers
106 views

$\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$

Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup ...
1
vote
1answer
152 views

Algorithm for finding a basis of a subgroup of a finitely generated free abelian group

Let $G$ be a finitely generated free abelian group. Let $\omega_1,\cdots, \omega_n$ be its basis. Suppose we are given explicitly a finite sequence of elements $\alpha_1,\cdots, \alpha_m$ of $G$ in ...
1
vote
0answers
153 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
0
votes
1answer
104 views

Let G be an abelian group and fix a prime p. Prove that G/P has no element of order p.

Let $G$ be an abelian group and fix a prime $p$. Let $P=\{g\in G|\operatorname{ord}(g) \text{ is a power of }p \}$. Let's assume that $P$ is a subgroup of $G$. Prove that $G/P$ has no element of ...
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votes
2answers
372 views

Give an example of a nonabelian group such that G/Z(G) is… [closed]

A) abelian; B) nonabelian; Not sure here.
0
votes
1answer
273 views
1
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1answer
794 views

If $Y: G\to H$ is a group homomorphism and $G$ is abelian, prove that $Y(G)$ is also abelian.

What I got: Suppose $Y$ is a homomorphism and that $G$ is abelian. Then for all $a,b \in G$, $ab=ba$, and thus $Y(ab)=Y(ba)=Y(a)Y(b)=Y(b)Y(a)$. However, this seems too simple and I was confused on ...
5
votes
1answer
98 views

What is the inverse limit of $…\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}$ (multiplying by all positive integers)?

According to a modified answer of this question, the direct limit of the sequence $$ \mathbb{Z}\xrightarrow{1}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{4}... $$ in the ...
0
votes
1answer
654 views

Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.

it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these http://en.wikipedia.org/wiki/Hamiltonian_group and saw the quaternion group, ...
1
vote
0answers
141 views

Specific basis of a subgroup of a free abelian group

I'm looking for clarification on Fraleigh's "A First Course in Abstract Algebra" Theorem 38.11. It states: "Let $G$ be a nonzero free abelian group of finite rank $n$, and let $K$ be a nonzero group ...
5
votes
2answers
446 views

Group isomorphism: $\mathbb{R}/\mathbb{Z}\cong S^1$

Let $\mathbb R$, $\mathbb{Z}$ be the groups of real numbers and integers respectively under addition, and $S^1$ denote the group of complex with modulus $1$ under multiplication. Then show that ...
1
vote
2answers
89 views

$\text{order }a\otimes b=\gcd\left(\text{order }a,\text{order }b\right)$?

Let $A,B$ be abelian groups with $a\in A$ having order $m$ and $b\in B$ having order $n$. Then $m\left(a\otimes b\right)=\left(ma\right)\otimes b=0\otimes b=0$ and likewise $n\left(a\otimes ...
1
vote
1answer
181 views

Cyclic group of order $6$ defined by generators $a^2=b^3=a^{-1}b^{-1}ab=e$.

The cyclic group of order $6$ is the group defined by the generators $a,b$ and relations $a^2=b^3=a^{-1}b^{-1}ab=e.$ For this problem I am assuming that it must be $a,b \neq e$. I will show the ...
5
votes
2answers
150 views

Order of a product in an abelian group.

Suppose $G$ is a finite abelian group and has two element $a$ and $b$, such that $\circ(a)=m$ and $\circ(b)=n$ and $lcm(m,n)\neq m,n$. Is it true that $\circ(ab)=lcm(m,n)$? Thanks in advance.
0
votes
3answers
505 views

Prove or disprove that there is an abelian, noncyclic group of order 52.

So I've heard one must invoke Sylow's theorems in order to break down something like this. So far I know that there is a subgroup of order 13 in G, and that it's the only subgroup of order 13 in G. To ...
2
votes
2answers
146 views

Fundamental Theorem of Finite Abelian Groups

Fundamental Theorem of Finite Abelian Groups indicates that $\mathbb{Z}_{n}$ is isomorphic to $\mathbb{Z}_{p_1^{k_1}} \times \mathbb{Z}_{p_2^{k_2}}\times$ ... $\times\mathbb{Z}_{p_n^{k_n}}$ where ...
0
votes
2answers
116 views

Question about Abelian group proof

I prove that if $G$ is Abelian group so if $a,b\in G$ has a finite order so $ab$ has a finite order to.. (Maybe later I'll upload here my proof to see of she is correct....) Now, I have to show that ...
0
votes
1answer
113 views

Set of negligible functions $\phi$ with $Im(\phi) \subseteq N$. What does it mean?

I am solving the following problem. If I have a subset $G$ of negligible functions defined as $G=\{\phi$ is a negligible function where $Im(\phi)\subseteq \mathbb{N}\}$, and defined $(G,\circ)$. I ...
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votes
1answer
40 views

is this operating procedure an Abelian Group?

I have to show if the following procedure gives a (Abelian) Group (G, *). $G = \{ \textrm{true}, \textrm{false} \}$ $a*b := ( a \leftrightarrow b)$ (which means that $a$ is $\textrm{true}$ if and ...
2
votes
0answers
75 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
4
votes
1answer
176 views

If $G, H, K$ are divisible abelian groups and $G \oplus H \cong G \oplus K$ then $H \cong K$

This is an exercise in Hungerford. But can somebody explain why is the following not a counter-example? Let $G$ be the direct sum of $|\mathbb{R}|$ copies of $\mathbb{Q}$. Let $K$ be the direct sum ...
6
votes
1answer
375 views

Manipulating quotients and direct sums for abelian groups

I'm studying Homology in Hatcher's Algebraic Topology. I feel that there is a gap in my group theory knowledge that is making me struggle with this chapter. In particular, the book (and material ...