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
votes
2answers
44 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 $G$ ...
2
votes
1answer
37 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
45 views
-5
votes
3answers
56 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 ...
10
votes
6answers
1k views

Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
4
votes
1answer
84 views

Let $G$ to be abelian group and $|G|=mn$ when $(m,n)=1$. $G_m=\{g\mid g^m=e\}$,$G_n=\{g\mid g^n=e\}$, prove isomorphism

I want to prove $ f:G_n\times G_m\rightarrow G$ when $f(g,h)=gh $ is an isomorphism. First of all I showed that $G_m,G_n$ are subgroups of $G$ (easy). Now I want to show that for every $ a,b, ...
3
votes
1answer
137 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
2
votes
2answers
59 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 ...
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} ...
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 ...
2
votes
0answers
327 views

Find order of given factor group

I'm trying to find the order of this factor group: $$(\mathbb Z_{12}\times\mathbb Z_{18}) / \langle (4,3)\rangle.$$ The order of the factor group is just the number of elements in it (aka the ...
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 ...
3
votes
3answers
91 views

Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
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
1answer
57 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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
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 ...
6
votes
2answers
2k views

Proving that a subgroup of a finitely generated abelian group is finitely generated

A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated. I would say that for a finitely generated abelian ...
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
0answers
166 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
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 ...
-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
167 views

There are no maximal $\mathbb{Z}$-submodules in $\mathbb{Q}$

Is it true that $\mathbb{Q}$ viewed as $\mathbb{Z}$-module (i.e. abelian group) has no maximal $\mathbb{Z}$-submodules? Why ?
-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.
10
votes
3answers
267 views

How to prove that a group with some properties is abelian?

Let $(G,.)$ be a group and $m,n\in\mathbb Z$ such that $\gcd(m,n)=1 $ and $$ \forall a,b \in G:a^mb^m=b^ma^m$$ $$\forall a,b \in G:a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian group ? Thanks ...
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
28 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
61 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 ...
5
votes
2answers
382 views

Show that any abelian transitive subgroup of $S_n$ has order $n$

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? Let $G$ be a an abelian transitive subgroup of the symmetric group $S_n$. Show that ...
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 $N$ ...
-4
votes
1answer
72 views

$\mathbb{R}$ is not a direct sum of its subgroups [closed]

How to prove the set of real numbers under addition; i.e., $(\mathbb{R}, +)$, is not the direct sum of two of its proper subgroups?
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
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 ...
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 $n \ge 3$ ?
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 ...
10
votes
1answer
117 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
1
vote
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
vote
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 ...
6
votes
2answers
58 views

Quotient group $\mathbb Z^n/\ \text{im}(A)$

Let $A$ be an $n \times n$ matrix with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n /\ \text{im}( \phi )$ (here $\phi : v \mapsto Av$ )? This problem ...
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 ...