Tagged Questions

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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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 $g^k = e$ ...
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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 $HK=KH$ $|HK|=O(H)O(K)/O(H \cup K)$ $l=(mn)/O(H \cap K)$ ...
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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 ...
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Properties of cyclic abelian group [on hold]

I am having trouble with this group theory question. Could some one solve it for me? Let $G$ be an abelian group, and let $a\in G$. For $n\geq1$, let $$G[n; a] = \{x\in G : x^n = a\}.$$ a. Show ...
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Orders of the elements in Z/8Z

I know that the order of an element a in a group is the smallest positive integer n such that a^n=1. enter image description here So for lets say order of an element 0 in the group Z/8Z, the order ...
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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.
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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 ...
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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 : ...
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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 ...
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Abelian groups as the commutator subgroup of groups [closed]

Let $H$ be an abelian group, is there a group $G$ such that the commutator subgroup of $G$ is isomorphic to $H$?
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Abelian group Of odd order problem [closed]

Prove for an Abelian group $G$ of order $(2n+1)$ with $n\in\mathbb{N}$, the products $gi gi$ for i=1,..,2n+1 cover the group $G$.
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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 ...
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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 ...
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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 ...
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$\text{Aut}(G) \cong \text{Out}(G)$ 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 ...
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$Aut(G)$ 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, ...
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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 ...
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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! ...
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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 ...
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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 ...
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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: $4/\gcd(3,4)=4$ ...
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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 ...
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Proving that the maximal abelian extension contains all abelian extensions

I have a small exercise I would like to get help with: Let $K$ be a field and $K^{ab}$ be the composite of all finite abelian extensions of $K$. Prove $K^{ab}/K$ is abelian and that if ...
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Where is the condition of infinite cyclic being used?

While reading Munkres' Topology, I came across this lemma. Let $G$ is a free abelian group with basis $\{g_{\alpha}\}$. If $H$ is any other abelian group and if $\{y_{\alpha}\}$ is a family of ...
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What exactly is a Frieze group and how would you find the isometries preserving one?

So far, I've come across several examples of frieze groups, but I've not yet come across an understandable definition of what they are. I've also been asked questions that ask me to state the ...
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primary decomposition of a group

Let G be the set of all pairs (x, y) of congruence classes x, y ∈ Z12 such that 4x + 6y = 0. Consider G as a subgroup of Z12 ⊕ Z12. Find the primary decomposition of G. I know that the primary ...
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for some prime number p, if for the group G, $|G| = p^2$ then either G is abelian or the centre of G = {e}

Told to use the face that if the quotient group of G and its centre is cyclic, then G is abelian. ie: $G \over Z(G)$ is cyclic $\Rightarrow$ G is abelian
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Commutator Subgroup in a $p$-group

Let $G$ be a finite non-trivial $p$-group. Show that $G'$ (the commutator subgroup of $G$), is a proper subgroup of $G$. How could one show this result? I was thinking of first arguing that the ...