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
22 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
-2
votes
1answer
43 views

Are there any examples of non-abelian subgroups of abelian group? [on hold]

Are there any examples of non-abelian subgroups of abelian group?
1
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0answers
20 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ we ...
-5
votes
3answers
55 views

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an abelian group [on hold]

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an Abelian group with the multiplication operation of complex numbers.
-3
votes
1answer
48 views

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? [on hold]

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? This question is from group theory in Abstract Algebra and ...
0
votes
0answers
10 views

A representative subspace for the cosets of another subspace $H$.

Suppose I have a subspace $H \leq V$ over a commutative ring (or if I can't get what I want with that generality, a field). I would like to specify a subspace $K$ such that: -$K \cap H = \{ \vec{0} ...
2
votes
2answers
58 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
3
votes
0answers
38 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
0
votes
2answers
14 views

Conjugate closure and factor group

Let $N\unlhd K$ be a normal subgroup of a given group $K$ and let $$q:K\to K/N$$ be the natural quotient map. Let $A\subseteq K$ be a subset of $K$ and let the conjugate closure of $A$ in $K$ be ...
3
votes
1answer
50 views

Prove the group is a direct product [closed]

Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
2
votes
1answer
61 views

Showing that the product $x*y := \frac{x+y}{xy+1}$ is a group operation on $(-1, 1)$ [duplicate]

I need to show that the following is an abelian group: $$x*y = \frac{x+y}{xy+1}$$ on the set $\{x \in \Bbb R \,|\, -1 < x < 1\}$. I have been working on this problem, trying to show ...
1
vote
1answer
23 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
0
votes
1answer
38 views

Prove that $C_8\times C_2$ has an isomorphic subgroup U and $G/U$ is isomorphic to $C_4$.

Let $G=C_{p^{k_1}}\times C_{p^{k_2}}\times ... \times C_{p^{k_n}}$ an abelian $p$ group, while $k_1,...,k_n\in\mathbb{N}$ and $k_1\geq k_2 \geq ...\geq k_n$. A group $U\cong C_{p^{l_1}}\times ...
2
votes
2answers
200 views

Number of elements in a finite abelian groups

Is the following true? Let $G$ be a finite abelian group with a minimal generating set $S$. By minimal generating set I mean we cannot reduce the cardinality further. Let $S=\{a_1,a_2,\ldots ,a_k\}$ ...
2
votes
1answer
38 views

Subgroups of Abelian Group of order 1000

Suppose you have an abelian group of size 1,000. How many subgroups does it have? I know there are 9 such groups from $1,000 = 2^3 \times 5^3$ giving us 3 of order $2^3 \times$ 3 of order $5^3$ ...
1
vote
4answers
37 views

Prove that $(G, *)$ is abelian group, for $ x * y = \tan^{-1}(tan(x) + tan(y))$

I have some troubles solving this problem. In order to prove that $(G, *)$ is an abelian group I have to find the identity element of the group, first; $\exists \ e \in \ G \ and \ x \in G$ such that ...
2
votes
1answer
38 views

Question about the assumptions to have $G \simeq H\times K$

I've been looking this fact: Let $G$ be a group, with $G$ abelian. Let $H$, $K \leq G$, with $G=HK$ and $H\cap K=\{e\}$. Then, we have that $G \simeq H\times K$. And my question is: We know ...
1
vote
1answer
46 views

If $a\otimes(b\otimes c)=0$ then $(a\otimes b)\otimes c=0$

I'm trying to prove the identity above, while the tensor product is between members of abelian groups $A,B,C$. This seemed trivial to me at first but since the tensor products are quotient groups I ...
2
votes
1answer
36 views

Subgroup rank-complement existence

Let $G$ be a finitely generated Abelian group. For each subgroup $H$ of $G$, does there exist another subgroup $K$ of $G$ such that $\text{rank}(G)=\text{rank}(H)+\text{rank}(K)$ and ...
1
vote
1answer
58 views

Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions

Prelude Cconsider the finite abelian group $\mathcal G = \prod_{i=1}^A \mathbb Z_{a_i}$ and let $\mathbf s \in \mathcal G$. Let $\mathcal H = \operatorname{grp}({\mathbf s})$ be the subgroup of ...
3
votes
1answer
75 views

Direct sum isomorphism

Sorry, this may not be a good question here but I have no idea. Let $\{A_{i}; i\in I\}$ and $\{B_{i}; i\in I\}$ be two different collection of abelian groups. The index set $I$ is the same in ...
2
votes
1answer
34 views

Solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$

How to solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$? The origin of this problem is the following question: Show that rank-2 free $\mathbb Z$ module $\mathbb Z^2$ has $p+1$ ...
0
votes
1answer
27 views

Irreducible characters of finite abelian groups

Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega_d$ be a primitive $d$-root of unity and ...
2
votes
1answer
60 views

Find the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n

What is the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n? This is a question in Joseph A.Gallian's book, and the answer is n=36 and the ...
-2
votes
2answers
80 views

what is the way to reach apropriate answer about my question about group in algebra [duplicate]

Assume there exist $2$ natural numbers that are coprime ($m$ and $n$ such that $(m,n)=1$) such that for each $g$, $h \in G$ we have $g^m h^m = h^m g^m$ and $g^n h^n = h^n g^n$. Then $G$ is abelian ...
-3
votes
1answer
96 views

Assume two natural numbers $m,n$ are coprime such that $a^m b^m=b^m a^m$ and $a^n b^n=b^n a^n$. Then $G$ is an abelian group [duplicate]

Assume two natural numbers $m,n$ are coprime (this means $(m,n)=1$) such that for each $a ,b \in G$ we have $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$. Then $G$ is an abelian group.
2
votes
2answers
55 views

Presentation of the additive group of the rational numbers

We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as $$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$ Then ...
3
votes
0answers
56 views

Rank of an abelian group

I learned that a rank of an abelian group is defined by a cardinality of maximal linearly independent sets. But how we can say that this is well-defined? I mean, I want to show that if $M$ and $N$ ...
1
vote
2answers
45 views

Objects that are quotient of two projective objects and cohomology in degree>1

1) What is an example of an abelian group which is not the quotient of two free abelian groups? For the abelian group $X$ for which this is true then for all Right exact functors F, i would have ...
1
vote
2answers
31 views

How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$?

How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$ ?
1
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2answers
55 views

Proving a quotient group is not Abelian without calculating actual cosets

Given the normal subgroup of S4: N={(1),(12)(34),(13)(24),(14)(23)}, show that S4/N is not Abelian. What I did was to calculate two random cosets of N in S4,like in the picture I attached, and show ...
1
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3answers
43 views

The center of a non-Abelian group of order 8

Let G be a non-Abelian group of order 8. Prove that $|Z(G)|\leq2$. (The center $Z(G)$ is defined as $Z(G)=\{ a\in G | ag=ga$ for all $g\in G \}$). I deduced from Lagrange's theorem that ...
0
votes
1answer
31 views

$B^n=A^n \cap B$ for every natural $n$. If $A/B$ is finitely generated, then $B$ is a direct factor of $A$

Let $A$ be an abelian group, and $B \le A$. Suppose that $B^n=A^n \cap B$ for every natural $n$. Prove that if $A/B$ is finitely generated, then $B$ is a direct factor of $A$. Notation: Let $G$ ...
2
votes
1answer
43 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
2
votes
1answer
30 views

The abelian group of smallest order and smallest non prime integer n divides |G| but G doesn't have an element of order n?

I don't know how to think of an example. What's an example of such a group. It doesn't make sense to me because if it is a finite abelian group, it can be written as a direct product of the integers ...
0
votes
2answers
80 views

Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
1
vote
1answer
23 views

How do I check this simple set is an Abelian group?

The n-gon in question is a 3-gon. It is an equilateral triangle to be exact. This is a Dihedral group of order 6 (3 reflections and 3 rotations) I have plotted the Cayley's table. The set of ...
1
vote
1answer
56 views

Is this true for finite abelian groups?

I'm trying to decompose abelian groups using the structure theorem for modules over PIDs. Here is one I have proved: Let $G$ be a non-trivial abelian group whose prime decomposition of the order ...
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2answers
86 views

Is every subgroup of a group normal?

Is there a simple example that can be used to show that not every subgroup of a group is normal? thanks,
1
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5answers
119 views

Prove every group of order less or equal to five is abelian [closed]

Is it possible to prove that every group of order less or equal to five is abelian? thanks,
0
votes
0answers
23 views

Looking for an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$?

Please do we have an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$? For example in $Q_8$ which is non-abelian, all the elements in ...
1
vote
2answers
25 views

Locally graded group with all proper subgroups abelian

A group $G$ is said to be locally graded if every finitely generated nontrivial subgroup of $G$ contains a proper subgroup of finite index. I have to prove that a locally graded group with all proper ...
0
votes
1answer
63 views

if o(a) is equal to exponent of finite abelian group G then $G=<a>\times K$

problem:prove that if $o(a)$ is equal to the exponent of a finite Abelian group $G$, then there exists $H<G$ such that $G=H\times\langle a\rangle$$$$$ using fundamental theorem of finitely ...
5
votes
2answers
51 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
4
votes
1answer
88 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
8
votes
1answer
140 views

A problem about abelian group

Given a group $G$, let $G_m$ be the group generated by the set $S=\{g^m|g\in G\}$. Prove that if $G_m$ and $G_n$ are both abelian, then $G_{\gcd(m,n)}$ is also abelian.
1
vote
2answers
68 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
0
votes
1answer
31 views

Conditions on subgroups $H, K$ of an abelian group $G$ such that $G/K \cong H/(H \cap K)$

I am trying to prove the equivalence of two formulations of simplicial homology on a manifold $X$, both of which are defined as the quotient of a certain set of simplicial chains on $X$ by a certain ...
0
votes
1answer
45 views

Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$.

I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look ...
4
votes
1answer
70 views

Sum of elements of a finite Field

Let $F$ be a finite field and $i$ an integer. Calculate the sum of all the elements of $F$,each raised to the $i-th$ power. My approach so ...