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

1
vote
0answers
33 views

Abelian Groups and orders

I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was: Let $G$ be an abelian ...
7
votes
1answer
66 views

Does there exist an $n$ such that all groups of order $n$ are Abelian?

I know that all groups of order $\leq$ 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?
1
vote
2answers
15 views

Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
2
votes
3answers
57 views

Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
1
vote
0answers
41 views

Difficulty understanding subgroups for certain simple groups

Firstly, I am studying at the high-school level so please excuse my lack of understanding of these concepts. Consider the group $\Bbb C_{3v}$ of symmetries of an equilateral triangular lamina. It ...
1
vote
0answers
30 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
2
votes
1answer
38 views

Number of group homomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$

I know that there does not exist an isomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$ as there exist an element of order 8 in $Z_8$ ⊕ $Z_2$ and no element of order 8 in $Z_4$ ⊕ $Z_4$. But what about ...
0
votes
1answer
45 views

Proving that a homomorphism is abelian [duplicate]

The question states... Let $\phi: G \to G$ be a homomorphism with the map $\phi(g)=g^2$. Prove $\phi$ is abelian. So far I have: Let $g$ and $h$ be in $G$. Then $\phi(g)=g^2$ and $\phi(h)=h^2$. ...
1
vote
0answers
30 views

Order of elements in certain generating sets of non-abelian groups !!!

The following example is just to clarify the idea. Example: The dihedral group has the following presentation $$D_{2i}=\left<s,r/s^2=r^i=e,sr=r^{-1}s \right>$$ Let $S_1=\{s,r\}$, ...
0
votes
1answer
52 views

Subgroups of abelian-by-finite groups

I am trying to prove that a subgroup of a abelian-by-finite group is also abelian-by-finite. I am not sure if I can use the same procedure that is used for a subgroup of a ...
-3
votes
1answer
56 views

$f:A\to B$ epimorphism if and only if $B/f(A)$ is a torsion module [closed]

I want to prove the double implication: $f: A\to B$ is an epimorphism in the category of torsion-free abelian groups $\Longleftrightarrow$ $B/f(A)$ is a torsion $\mathbb Z$-module. Thanks.
1
vote
1answer
43 views

Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
3
votes
0answers
48 views

Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
0
votes
1answer
66 views

Isomorphism between $A$ and $B$ if $A/kA$ and $B/kB$ have same order? [closed]

Let $A$ and $B$ be finitely generated abelian groups, where for all $k\geq1$ the orders of $A/kA$ and $B/kB$ are equal. Are $A$ and $B$ necessarily isomorphic? Why (not)?
0
votes
1answer
81 views

Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...
3
votes
1answer
87 views

Finding an explicit isomorphism from $\mathbb{Z}^{4}/H$ to $\mathbb{Z} \oplus \mathbb{Z}/18\mathbb{Z}$

There was a past qualifying exam problem, I was having trouble with, it is stated below as follows: In the group $G= \mathbb{Z} \times \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}=\mathbb{Z}^{4}$, ...
1
vote
0answers
32 views

Proving property of cyclic groups

A user asked the following question. It was closed as off-topic, or rather as missing context, but it seems the context close reason doesn't exist, so off-topic was chosen. Here it is: I am having ...
3
votes
1answer
49 views

Exercises Involving Torsion and Abelian Groups

I am working on the following group-theory exercises but I'm a little confused by how to begin proving them. Let $G$ be a group. Call $g \in G$ a $torsion \ element$ if $g$ has finite order $g^k = e$ ...
1
vote
1answer
48 views

o(a)=m ,o (b) =n , ab=ba then o(ab)=lcm(m,n). What happens when b is the inverse of a?

Let $G$ be a group and let $a,b \in G$ s.t $O(a)=m$ and $O(b)=n$ and $ab=ba$. Then $O(ab)=lcm( m,n)$. My attempt: since $ab=ba$ then $HK=KH$ $ |HK|=O(H)O(K)/O(H \cup K)$ $l=(mn)/O(H \cap K)$ ...
6
votes
1answer
98 views

Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
0
votes
1answer
75 views

Orders of the elements in $\mathbb{Z}/8\mathbb{Z}$

I know that the order of an element $a$ in a group is the smallest positive integer $n$ such that $a^n = 1$. You know $\mathbb{Z}/8\mathbb{Z} = \{\overline{0}, \overline{1}, \dotsc, ...
1
vote
0answers
56 views

Which lines in $\mathbb R^2$ define a subgroup?

Which lines in $\mathbb R^2$ define a subgroup? I know that the line $y=x$ in $\mathbb{R}^2$ gives a subgroup, but I can't figure out the other ones.
0
votes
0answers
32 views

Find two groups that give a cyclic decomposition of $G=\mathbb{Z}_4 \oplus \mathbb{Z}_2$

Let $G=\mathbb{Z}_4 \oplus \mathbb{Z}_2$. Find two distinct pairs of subgroups of G such that each pair gives a cyclic decomposition of G with no subgroup of the second pair equal to either subgroup ...
-3
votes
1answer
58 views

Is this construction algebraically closed?

On the tetration forum Tommy1729 proposed a new kind of number : http://math.eretrandre.org/tetrationforum/showthread.php?tid=1036 Too avoid deletion or changes of that post , I copy it here : ...
1
vote
2answers
57 views

Commutative free products

Do there exist any non-trivial groups such that their free product is commutative? That is, if $G, H$ are non-trivial groups is $G*H$ ever commutative? My thinking is no but I can't really formulate ...
0
votes
0answers
45 views

Questions related to semidirect-product of Klein four group?

I have four questions related to Klein four group. and I know the answer two of them. ( the first two) and I want to know answer Does $V_4$ has an automorphism of order 6? What are the orders of ...
0
votes
1answer
28 views

Algebraic structures [closed]

I can't wrap my head around this area in mathematics. What is a group, a, semigroup, what is a field, a ring, an abelian group? I read all sorts of texts, but it's so abstract. I can't solve problems ...
1
vote
1answer
40 views

How to construct an abelian permutation group?

I am studying binary operations and permutation groups. One of the exercices leaves me a bit perplex, that is: "Construct a permutation group which is Abelian(commutative)" From what I understand, a ...
0
votes
1answer
48 views

$\text{Aut}(G) \cong \text{Out}(G)$ when $G$ is abelian

I would appreciate if you could please evaluate my proof and point out any mistakes I made. Proof: Define a homomorphism $\Phi: \text{Aut}(G)\to \text{Out}(G)$, such that all elements of ...
5
votes
2answers
132 views

$Aut(G)$ is cyclic $\implies G$ is abelian

I would appreciate if you could please express your opinion about my proof. I'm not yet very good with automorphisms, so I'm trying to make sure my proofs are OK. Proof: Since $Aut(G)$ is cyclic, ...
1
vote
0answers
15 views

Finite abelian group of orthogonal matrices

Let $G$ be a finite abelian subgroup of $O_{2n+1}(\mathbb R)$. Suppose that $G\not\subset \{-I_n,I_n\}$. There exists $\varepsilon \in \{-1,1\}$ and $M\in G$ such that $0<\dim \ker (M-\varepsilon ...
0
votes
0answers
14 views

Help with Abelian group and homomorphisms

Let A be an abelian group and k∈Z. a) Show that hk:A→A defined by hk (a)=ka is a homomorphism. b) In the case A= Z , show that any homomorphism Z→Z must be hk for some k . Any help ...
0
votes
2answers
31 views

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1 }$.

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1}$. I know that I should show that $X*Y=Y*X=e$. But I don't know how to calculate it.
1
vote
1answer
15 views

Subquotient of abelian group equivalent to “quotientsub” of abelian group?

Suppose we have an abelian group $G$. Suppose also that we have a "subquotient" $H$, which is a subgroup of a quotient group of $G$. If $H$ can be constructed in this way, when is it also true that ...
2
votes
0answers
28 views

Group G with $\left(\forall g \in G: g^2=1_G\right) \implies \text{G is abelian}$ [duplicate]

I have the following task: Be $G$ a group with $\forall g \in G: g^2=1_G$ Prove that $G$ is abelian. I proved it this way: $\forall g \in G: g^2=1_G$ implies $\forall g \in G: g=g^{-1}$ ...
1
vote
1answer
125 views

Let G be an abelian group, and let a∈G. For n≥1,let G[n;a] := {x∈G:x^n =a}. Show that G[n; a] is either empty or equal to αG[n] := {αg : g ∈ G[n]}… [closed]

We were given questions to study for our exam coming up. We have not covered much of this topic, so any help would be greatly appreciated! Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, ...
0
votes
1answer
9 views

On ordered abelian groups containing $\mathbb{Z}$

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$. I have no ...
3
votes
1answer
41 views

Commutativity of a Lie algebra $\Rightarrow$ the Lie group is abelian

Let $G$ be a Lie group, $\mathfrak{g}$ it's Lie algebra. Assume $[x,y]=0 \, \, \forall x,y \in \mathfrak{g}$. Is it true that $G$ is abelian? Remarks: (1) The other direction ($G$ abelian ...
1
vote
3answers
63 views

$H=\{x^2:x\in G\}$ then $H\unlhd G$

We have $H<G$ and element of $H$ is of the form $x^2$ where $x \in G$. H is a normal subgroup of $G$. The factor group $G/H$ is abelian. I tried first one by showing that $gh^2g^{-1} \in H$. ...
1
vote
2answers
26 views

Find the order of a subgroup of $\mathbb C^\times$

Let $\mathbb{C}^\times$ the multiplicative group of complex numbers different of zero. Let $H$ the subgroup $\mathbb{C}^\times$ of generated by $\{i, e^{\frac{2i \pi}{5}}, -1\}$. Find the order of ...
0
votes
1answer
45 views

Why is $\Bbb{Z_2} / \{e\} = \Bbb{Z_2}$?

Let the group $\Bbb{Z_2} = \{e, a\}$. We are given the quotient group $\Bbb{Z_2} / \{e\}$. So this gives us a set of left cosets: $\Bbb{Z_2} / \{e\} = \{e\{e\}, a\{e\}\} = \{\{e\}, \{a\}\} \neq \{e, ...
1
vote
0answers
50 views

Number of isomorphism classes of abelian groups of any order

Let $N$ be the order of an abelian group. The prime factorization is given by $N=\prod_{i=1}^{n}p_{i}^{e_{i}}$ with $p_{1}< p_{2}< \dots <p_{n}$ and $e_{i}\geq 1$. Let $\pi(n)$ denotes the ...
2
votes
4answers
306 views

Prove that a group is cyclic [closed]

$G$ is abelian of order $35$. and for all $x\in G$, $x^{35}=e$. I need to show that $G$ is cyclic. This seems perfectly obvious but I dont know how to write the proof. Help would be appreciated! ...
0
votes
2answers
69 views

How can you tell whether two groups are homomorphic/isomorphic? [closed]

Suppose you have two groups, $G$ and $H$. I've been taught the following definitions: "$G$ is homomorphic to $H$ iff there exists some function $\theta$ which gives the mapping $\theta : G ...
5
votes
2answers
276 views

Express an abelian group given as finite generators and their relations as a direct sum of cyclic groups and find corresponding generators.

According to page 158 of Dummit and Foote's Abstract Algebra (3rd edition): Theorem. (Fundamental Theorem of Finitely Generated Abelian Groups) Let $G$ be a finitely generated abelian group. Then ...
2
votes
0answers
40 views

Commutator subgroup is the minimal normal subgroup such that quotient group is abelian

I was recently asked this in Abstract Algebra class on group theory: Let G be a group and G' its commutator subgroup (i.e. its minimal subgroup containing all commutators $ [x,y] = xyx^{-1}y^{-1} ...
2
votes
3answers
79 views

Show that $a^{m} a^{n} = a^{m+n}$

Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that $$a^{m} a^{n} = a^{m+n}.$$ I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when ...
6
votes
0answers
184 views

Solving particular type of system of equations in $\mathbb R/\mathbb Z$

I apologize in advance for the long post. You can freely skip to the last paragraph. I was motivated by this question given in 5th grade mathematics competition that I was solving with my advanced ...
1
vote
0answers
56 views

The order of an element in a direct sum

I have a quick question: Say I have $\mathbb{Z}_4 \oplus \mathbb{Z}_5$ and I want to find the order of an element, say $(3,4)$. Well, in $\mathbb{Z}_4$, the order of 3 is equal to: $4/\gcd(3,4)=4$ ...
0
votes
1answer
30 views

Group direct sum construction. Problem from Undergraduate Algebra by Serge Lang.

I have been reading on Undergraduate Algebra by Serge Lang, and not so far deep in I got myself stuck already. The problem says: If $A$ is an abelian group, written additively, and $n$ is a ...