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
22 views

List all abelian groups that have order 81 and contain an element of order 27

List all abelian groups that have order 81 and contain an element of order 27. For each, give the primary decomposition and a specific element having order 27. I know $81 = 3^{4}$ so the abelian ...
0
votes
0answers
12 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 ...
0
votes
2answers
28 views

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian.

Give a specific example to show that $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I know that $S_{4}$ is not abelian and therefore $\mathbb Z_{2}$ × $S_{4}$ is not abelian. I'm not sure how to show this ...
4
votes
2answers
111 views

On subgroups of abelian groups

Let $G$ be a product of $n$ finite cyclic groups. Is every subgroup of $G$ also a product of (at most) $n$ finite cyclic groups ? I do not know the answer to this question, but I'm tempted to say ...
4
votes
1answer
33 views

Determining the multiplicative group of a ring of polynomials

Let us say that we have the polynomial ring R[x]. Would it be possible to determine the order of the multiplicative group of R[x] modulo a polynomial f?
2
votes
1answer
35 views

If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
0
votes
1answer
21 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
0
votes
0answers
22 views

homorphisms of abelian groups

Please help me with resolving this problem from Romanian "Gazeta Matematica": "a finite Abelian group $G$ such that $|\text{End } G |$ and $|\text{Aut } G |$ are coprime numbers. Show that $G$ is ...
0
votes
1answer
14 views

Equivalence of definitions of injective modules

Wikipedia article gives a number of definitions of injective modules, namely: If $Q$ is a submodule of some other left $R$-module $M$, then there exists another submodule $K$ of $M$ such that $M$ is ...
1
vote
1answer
23 views

Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
0
votes
2answers
51 views

Example of an infinite abelian group having a non-cyclic finite subgroup [closed]

Give example (if exists) of an infinite abelian group having a non-cyclic finite subgroup . Please help
8
votes
1answer
150 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
1
vote
2answers
30 views

Abelian group, inverse element

What would be the inverse element for this abelian group $[1,2,3,4,...,p-1]$ , in which p is a prime number, with this operation $(a*b)mod.p$? For all $a$ and $b$ of the set. I know the inverse ...
1
vote
1answer
35 views

Center of an abelian group

Prove if $G$ is non abelian group, then exists an abelian subgroup $H$ which contains $Z(G)$ and $H≠Z(G)$.
-1
votes
0answers
36 views

Finitely generated metabelian group

Let $G$ be infinite finitely generated metabelian group (i.e., there exists a normal abelian subgroup $H$ such that $G/H$ is abelian). I want to prove that $H$ is a torsion-free group. Is it true? ...
0
votes
1answer
17 views

Inverse element of the abelian group (P(M),symmetric difference)

What would be the inverse element in this abelian group: $(P(M),\triangle)$? I know the neutral element is the empty set and I thought the inverse element would be $A^{c}$ for every $A$. Turns out ...
0
votes
1answer
30 views

Why homomorph and not isomorph?

Why are the groups $\mathbb{R},+$ and $\mathbb{R}_0^+,*$ homomorph, their mapping function being $ f: x \rightarrow e^x $? Why is this not an isomorphism?
1
vote
1answer
44 views

finetely generated Abelien -by-nilpotent group

Let G finetely generated Abelien -by-nilpotent group (i.e there existe a abelien subgroup H in G and G/H is nilpotent )With each of its two-generator is nilpotent-by-finite show that G is ...
-3
votes
1answer
64 views

Show there is no non-abelian group of order 9 [duplicate]

I want to show there is no non-abelian group of order 9. How should I attempt this?
0
votes
2answers
48 views

Non abelian subgroup of a abelian group.

What is the relationship between abelian subgroup of a non-abelian group(when exist, example, theorem)?? any thing such link regarding the question would help. ...
1
vote
2answers
26 views

Order and Least Common Multiple Abelian Question

\item Let $G$ be an abelian group and let $x, y\in G$ be elements so that $o(x)=m$ and $o(y)=n$. Show that $o(xy)=\frac{mn}{(m,n)}$. (Note that this is the least common multiple of $m$ and $n$) Is ...
2
votes
1answer
54 views

How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is ...
1
vote
1answer
22 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
1
vote
1answer
21 views

$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
1
vote
5answers
56 views

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? [duplicate]

If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well? Since $(ab)^n=e$, I get $(b)(ab)^n(a)= ba$. This means that $(ba)^{n+1}=ba$, and $(ba)^n=e$. But, I ...
1
vote
0answers
47 views

Let $G$ be a group and let $\Phi\colon G \to G$ be an isomorphism. Define $H = \{ a \in G\ |\ \Phi(a) = a^{-1} \}$ [duplicate]

It asks to prove that if $H$ is a subgroup of $G$, then $G$ is abelian. Solution: I showed that for every $a$ and $b$ in $H$, $a$ and $b$ commute. But how do I generalize to elements in $G$ NOT in ...
1
vote
2answers
47 views

Prove rk$B$ $\le$ rk$A$ where A and B are free, abelian and finitely generated groups.

Let $A$ and $B$ be free abelian, finitely generated groups. Let $f:A \to B$ be an epimorphism. Prove rk$B$ $\le$ rk$A$. I could really use a verification. That is a question from my exam today. ...
2
votes
0answers
56 views

$gcd(|G|, |Aut(G)|)=1$ means G is abelian?

Prove the following assuming that G is finite group with $gcd(|G|, |Aut(G)|)=1$ a)G is abelian (done) b) Every Sylow subgroup of G is cyclic of prime order. G is abelian than every sylow unique, ...
0
votes
1answer
46 views

Prove that $G$ has a subgroup $L$ and $|L|=mn$

For $G$ is an abelian group, $H,K$ are subgroups of $G$ and $|H|=n,|K|=m$. Prove that $G$ has a subgroup $L$ and $|L|=mn$ In cases $H\cap K=\{e\}$ use Lagrange theorem we can show that $|HK|=mn$ but ...
6
votes
0answers
70 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
-1
votes
2answers
45 views

The generator of a cyclic quotient group

(I'll use (a) to denote the subgroup of G generated by a) Let G be a finite abelian group of order n. Let a be an element of G of order k. We can easily see that (a) is a normal subgroup of G. If we ...
0
votes
2answers
34 views

Prove $G\cong H\oplus \Bbb{Z}^{k}$.

Let $G$ be an abelian group and let $H$ be a subgroup. Let $G/H\cong \Bbb{Z}^{k}$. Prove $G\cong H\oplus \Bbb{Z}^{k}$. What I did so far is: there is an epimorphism from $G$ to $\Bbb{Z}^{k}$ such ...
0
votes
1answer
32 views

Questions in Abstract Algebra

I have two question which I couldn't solve: Let $G$ be a group of size $40$. a. Show the $5$-Sylow subgroup in $G$ is Normal - this part was easy, I just showed that $n5=1$ and then $P5$ is ...
3
votes
1answer
32 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
4
votes
0answers
35 views

Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
0
votes
2answers
25 views

The abelianness of the quotient group of an abelian group.

I am working on an assignment for my abstract algebra class. The question states: Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A/B$ is abelian. I was under the ...
2
votes
2answers
36 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
3
votes
2answers
85 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
2
votes
2answers
54 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
3
votes
2answers
89 views

Does there exist an abelian group with insoluable word problem?

Does there exist an abelian group with recursively enumerable presentation and insoluble word problem? My gut says "of course not!". However, my mind keeps saying "but...doesn't $\mathbb{R}$ have ...
4
votes
2answers
30 views

Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
1
vote
1answer
26 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...
0
votes
0answers
13 views

if A/B is a torsion group then rank(A)=rank(B)

Let $A$ a finitely generated abelian group, $B\subset A$ a subgroup such that $A/B$ is a torsion group. Then $rank(A)=rank(B)$
3
votes
1answer
61 views

Is the group $G =\{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ cyclic?

$G = \{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ under addition: I am going to say it's not cyclic because a,b can be distinct. I tried finding a generator.
1
vote
2answers
57 views

Law of Exponents for Abelian Groups

Let $a$ and $b$ be elements of an Abelian group and let $n$ be any positive integer. Show that $(ab)^n = a^nb^n$. Is this also true for non-Abelian groups?
3
votes
3answers
73 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 ...
1
vote
1answer
70 views

Group theory- rank of a group. What am I doing wrong?

I was given a question: Let $n\in \mathbb{N}$ and let $A$ and $B$ groups, both isomorphic to $\mathbb{Z}^n$. Let $f:A \to B$ be a surjective homomorphism. Prove $f$ is an isomorphism. Here's my ...
5
votes
0answers
41 views

Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
0
votes
0answers
40 views

Proving that product of two quotients = a certain quotient group

Question I am producing certain given conditions from a paper and a certain fact(stated in the paper) that I need to prove using those conditions. I am converting this problem into a general group ...
4
votes
1answer
48 views

If $A\oplus B\cong A\oplus C$ then $B\cong C$

Let $A,B,C$ be finitely generated abelian groups, and $A\oplus B\cong A\oplus C$. Prove that $B\cong C$. I know that it follows from the fundamental theorem of finite abelian groups, but I have ...