Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

learn more… | top users | synonyms

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
207 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
48 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
48 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 ...
1
vote
1answer
49 views

Rank-complement subgroup 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
62 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
76 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
35 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
32 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
76 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
82 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
102 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
57 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
58 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
48 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
vote
2answers
56 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
46 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
32 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
45 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
33 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
81 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
24 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
57 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 ...
-2
votes
2answers
88 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
vote
5answers
219 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,
1
vote
2answers
27 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
66 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
60 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
92 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
146 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
76 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
32 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
51 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 ...
5
votes
1answer
103 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 ...
1
vote
1answer
33 views

Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
3
votes
0answers
102 views

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
2
votes
3answers
130 views

For any element $g$ of $G,$ where $g$ has order $2,$ define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a subgroup of $G.$

Does this solution make sense? Let $G$ be an abelian group and $H$ a subgroup. For any element $g$ of $G,$ where $g$ has order $2$, define $gH=\{gh│h∈H\}$. Prove that the set $K=H∪gH$ is a ...
4
votes
1answer
44 views

Finding a subgroup of an abelian group that is isomorphic to Z

The question: If G is an abelian group and f is a surjective homomorphism from G to Z with kernel K, prove that G has a subgroup H such that H is isomorphic to Z. By the first isomorphism theorem I ...
0
votes
1answer
45 views

Relation between finite Abelian Groups and traces?

I have recently read Kronecker's 1870 paper on finite Abelian groups, on the definition of abstract group and so on. It turns out that such definition is literally taken over (being probably unaware ...
0
votes
1answer
49 views

Algebraic groups?

I have been doing group theory lately but I can not seem to find what I am looking for online (partly because I am not entirely sure what I am looking for). An example of one of the questions: If ...
1
vote
1answer
63 views

Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
2
votes
1answer
68 views

Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
1
vote
0answers
35 views

Invariant factors of a subgroup of a subgroup of $\mathbb{Z}^2$

Consider groups $B\leq A\leq\mathbb{Z}^2$. We have: A basis $e_1,e_2$ of $\mathbb{Z}^2$ and integers $a_1,a_2$ such that $a_1e_1,a_2e_2$ is a basis for $A$, and $a_1\mid a_2$. A basis $f_1,f_2$ of ...
0
votes
0answers
43 views

Direct sum of Abelian groups and Isomorphism

I'm currently reviewing my algebra for my last prelim and came across the following problem that has me stumped: If $A,B,C $ are finite Abelian groups such that $A\oplus B \cong A\oplus C$ then show ...
1
vote
1answer
27 views

Homomorphisms preserve solubility of groups, and some others.

Definition: Let $G$ be a group. A subnormal series for G is a chain of subgroups $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq G_n = G$ such that $G_i$ is normal in $G_{i+1}$ for $i = 0,1, ..., ...
2
votes
2answers
53 views

If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
1
vote
1answer
84 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...