Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$
0
votes
3answers
40 views
Given two sets, finding two non trivial homomorphisms that are not isomorphisms
Is it possible to have two non trivial homomorphisms that are not isomorphisms for given two Groups?
I am specially interested in additive/remainder Group of Integers and multiplicative (arithmetic ...
3
votes
1answer
31 views
Example of a group which is abelian and has finite (except the $e$) and infinite order elements.
Exercise 7: Show that the elements of finite order in an abelian
group $G$ form a subgroup of $G$
I just solved this exercise but I can't find example of a group which is abelian and has ...
1
vote
2answers
58 views
Finding invariant factors of finitely generated Abelian group
There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe).
Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
3
votes
1answer
74 views
Cocartesian squares in the category of abelian groups.
Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows.
Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
2
votes
1answer
49 views
Subgroups of $\mathbb{Z}^k$ of finite index $n$
I want to describe all subgroups in $\mathbb{Z}^k$ of finite index $n$.
I have solved it for the case $k=2$. In $\mathbb{Z}^2$, each subgroup of index $n$ corresponds to a matrix $\left( ...
4
votes
1answer
65 views
When are groups of order 12 non-abelian?
I am currently reading http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/group12.pdf, and have a quick question about the group being non-abelian. Let me explain:
Let $|G|=12=2^2\cdot 3$ and let ...
8
votes
1answer
60 views
Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?
What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime?
Obviously, each cyclic subgroup is generated by some ...
0
votes
1answer
62 views
4
votes
2answers
60 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
3
votes
1answer
47 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
3
votes
1answer
53 views
The group $\mathbb Q^*$ as a direct product/sum
Does the group $\mathbb Q^*$ (rationals without $0$ under multiplication) is a direct product or a direct sum of nontrivial subgroups?
My thoughts. Consider subgroups $\langle p\rangle=\{p^k\mid k\in ...
2
votes
1answer
97 views
Invariants of a subgroup.
I've been struggling many hours trying to solve this problem from the book: Topics in Algebra, of Herstein (2nd edition). If anybody can give me a hint for the solution, I would really appreciate ...
3
votes
2answers
46 views
Symmetric Groups and Commutativity
I just finished my homework which involved, among many things, the following question:
Let $S_{3}$ be the symmetric group $\{1,2,3\}$. Determine the number of elements that commute with (23).
Now, ...
1
vote
0answers
56 views
Free and torsion Group
Can you please explain torsion subgroup and free subgroup of free abelian group?
and also if $G$ is a finitely generated abelian group;
how is $G$ a direct product of free part and torsion part?
...
1
vote
0answers
72 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
$$
...
2
votes
2answers
101 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
3
votes
2answers
78 views
Abelization of a group is infinite cyclic
Suppose i have a group $G$ with the following presentation:
$$G= \langle a,b:a^p=b^q \rangle$$ with $p$ and $q$ coprime. I want to conclude that the abelization $G_{ab}$ infinite cyclic is. I have ...
2
votes
0answers
22 views
Separable elements of a finite abelian group
Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively.
...
0
votes
1answer
45 views
Quotient by a torsion group
Let $A$ be a finitely generated abelian group of rank $r$. The rank of the abelian group $A$ is the number of copies of $\mathbb Z$. Let $T$ be the torsion subgroup of $A$. Show that ...
3
votes
2answers
40 views
Show that the normal subgroup is cyclic
Let $G=\mathbb Z\times\mathbb Z$. Consider $H\leq G$ generated by $(-5,1)$ and $(1,-5)$. Show that $\frac{G}H$ is cyclic.
This is what I have so far but I'm not sure if I'm right either.
Let ...
1
vote
2answers
51 views
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian. [duplicate]
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$.
Show that $G$ is Abelian.
I need help on this problem. Appreciated!
3
votes
2answers
85 views
Number of Abelian Groups of Order 256
I am trying to find the number of abelian groups of order 256. Is the following correct?
We may write $256=2^8$ we then know that this may be represented in the form:
$C_{n_1}\times.....\times ...
2
votes
3answers
133 views
If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ .
If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ .
With the assumption, I dont know how to start the proof.
If there is no non-trivial ...
1
vote
3answers
90 views
Let $G=\mathbb Z_{10}\times\mathbb Z_{15}.$ How many elements of given orders?
Let $G=\mathbb Z_{10}\times\mathbb Z_{15}.$ Then which of the followings are correct:
$G$ contains exactly one element of order $2;$
$G$ contains exactly $5$ element of order $3;$
$G$ contains ...
0
votes
3answers
80 views
Does $a^n=b^n \implies a=b$ in an abelian group?
Does $a^n=b^n \implies a=b$ in an abelian group?
My intuition tells me this mug not always be the case. Under what conditions is this true?
Thanks!
3
votes
1answer
102 views
Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$
Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$.
I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
2
votes
0answers
69 views
$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$
Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
0
votes
2answers
56 views
Abelian Group Question: Why is $e^n e^m = e$
Let $G$ be an abelian group. Let $x,y \in G$. Let $m,n$ be positive integers. Assume that $x^m=e=y^n$. Prove that $(xy)^{mn}=e$ also.
So what I have done is:
$$
(xy)^{mn} = x^{mn} y^{mn} = (x^m)^n ...
0
votes
0answers
35 views
System of equations and Abel theorem
Consider this system of 3 equations to be solved in x,y and z:
$a x^m=(y+z)^n$
$by^m=(x+z)^n$
$cz^m=(x+y)^n$
The parameters $(a,b,c)$ and the unknown $(x,y,z)$ are all in $ℝ₊$. Also, m and n are ...
4
votes
3answers
55 views
Help finding all elements of order 2 in $S_6$.
I am trying to find all elements of order 2 in $S_6$. I am trying to understand how to achieve this. Here is my attempt.
We need only count the number of permutation of the forms
$
(a_1 a_2)\\
...
-3
votes
2answers
132 views
Homorphism Being Trival when Group Is Abelian
Let $G$ be a group, and let $g \in G$. The function $\gamma_g\colon G \to G$ defined by $(\forall a \in G)\colon \gamma_g(a)=g ag ^{-1} $ is an automorphism of $G$. The automorphisms $\gamma_g$ are ...
0
votes
3answers
58 views
Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian
Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
7
votes
1answer
81 views
If a finite translation of $A$ covers an abelian group, infinite translations of it intersect.
Let $G$ be an abelian group and $A\subseteq G$. Suppose there's a finite set $F\subseteq G$ such that:
$$G=FA$$
How can I prove any infinite translation of $A$ is overlapping, that is, there's not ...
7
votes
2answers
194 views
Abelian group admitting a surjective homomorphism onto an infinite cyclic group
I am working on the following problem:
Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective
homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$
By means of the First Isomorphism ...
0
votes
1answer
30 views
Definition of multiple sum
Suppose we have an abelian group $(G,+)$. What is the formal definition of multiple sums such as
$\sum_{i_1 \in A} \sum_{i_2 \in A_{i_1}} \cdots \sum_{i_n \in A_{i_{n-1}}}f(i_1,\ldots,i_n)$?
Thanks ...
1
vote
3answers
101 views
Why doesn't the Chinese remainder theorem contradict the Fundamental Theorem of Finitely Generated Abelian Groups?
I am finding a contradiction between those two theorems and I do not know what I am doing wrong. First theorem is:
The group $\Bbb Z_{m_1} \times \Bbb Z_{m_2} \times \dotsm \times \Bbb Z_{m_n}$ is ...
8
votes
1answer
93 views
on finite abelian groups
Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove
$$M(G)=\langle1\rangle \text{ or } Z_{2}$$
Attempt: We know that ...
1
vote
1answer
73 views
Can we determine structure the automorphism group of all infinite abelian groups? [duplicate]
Let $G$ be a infinite abelian group .
We know that we can determine structure the automorphisms group of all finite abelian groups.
Can we determine structuer the automorphisms group of all ...
1
vote
4answers
107 views
Proving that a set of matrices is an abelian group
Prove that the set of matrices in the form of
$\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha
&\cos \alpha \end{array}\right]$ (while $\alpha \in R$) with the
...
4
votes
1answer
40 views
Extending abelian groups to rings
I've been reading this article about extending abelian groups to rings: http://www.math.udel.edu/~coulter/papers/rings.pdf.
Could you explain to me why theorem 2.1 guarantees left and right ...
2
votes
3answers
91 views
Show that for G be a abelian group, $g \in G, g^m=1$, where $m=|G|$
I am pretty sure there is a name for the following theorem, but unfortunately, I don't known it.
Theorem. Let $G$ be a abelian group with $m=|G|$, than for any $g \in G, g^m=1$.
Currently I don't ...
4
votes
2answers
108 views
Determining whether two groups are isomorphic
I am reading "a first course in algebra" and there, i am trying to solve the exercises, but there is something i don't understand. How do we understand whether two groups are isomorphic or not? For ...
5
votes
1answer
104 views
Equivalences and isomorphisms of short exact sequences
In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
0
votes
2answers
86 views
About some bijections
Let $H=ℤ^{r},\ r>0$, and $K=ℤ/nℤ,\ n>0$. Let $G$ be an abelian group such that $H,K$ are subgroups of $G$ with $G=H+K$ and $H\cap K=\{0\}$. Then there is an isomorphism $\phi:H×K→G$ defined by:
...
3
votes
1answer
36 views
Maximal Subgroups Containing given Element
Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?
5
votes
5answers
126 views
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it?
I came across the above problem and do not know how to get it? Can someone point me ...
1
vote
5answers
154 views
Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$
Can you please help me in this question:
Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$.
Thanks a lot
1
vote
2answers
24 views
Enumerate Elements in Abelian Group
So I am reading in my book, and some across this example:
Consider the group $\mathbb{Z}^*_{15}$. We can enumerate its elements as:
$[\pm 1], [\pm 2], [\pm 4], [\pm 7]$
Can someone explain how the ...
6
votes
3answers
159 views
How to find subgroups of $ \;\;\Bbb Z_2\times \Bbb Z_6$
I am reading a first course in algebra and there is an example saying that "find all the subgroups of $\Bbb{Z}_2\times\Bbb{Z}_6$ and decide which of them are cyclic. I know that ...
5
votes
1answer
51 views
Additive category and zero map
Let $A$ be an additive category. Namely
$A$ has a zero object,
$A$ has finite products and coproducts, and
Every Hom-set is an Abelian group such that composition of morphisms is bilinear.
...
