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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13
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1answer
381 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
6
votes
1answer
247 views

Abelian subgroup in an infinite non-abelian $3$-group

Does an infinite non-abelian $3$-group has an infinite abelian subgroup? This result holds for $2$-groups, but I wonder whether this holds or not for $3$-groups. Thanks in advance.
4
votes
1answer
271 views

$G$ has exactly three subgroups

My attempt for the first: (I would like to get it verified because I didn't use property of a cyclic group) $|G|<\infty$ (since for otherwise $(a^2),(a^3)$ are distinct improper nontrivial ...
6
votes
1answer
72 views

Are there asymptotically more nonabelian groups of order $p^k$ than there are abelian groups of order $\leq p^k$?

Let $\alpha(n)$ denote the number of isomorphism classes of abelian groups of order $n$ and $\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define $f(p^k)$ to be the number of isomorphism ...
3
votes
1answer
202 views

Cauchy-Davenport theorem and its extension

According to Cauchy-Davenport Theorem, if $A,B$ are subsets of a prime field ($F_p$) then we have the following bound on the number of elements within the sumset $A + B = \left\{ {\left. {a + b} ...
4
votes
5answers
3k views

Is it true that a dihedral group is nonabelian?

Is it true that a dihedral group is nonabelian? I'm not sure if the result is true. I checked it for some lower order and I think the result may correct. But I failed to prove/disprove the result.
6
votes
2answers
206 views

Computing easy direct limit of groups

How do I start computing easy direct limit of groups: 1) $\mathbb{Z} \overset{1}\longrightarrow \mathbb{Z} \overset{2}\longrightarrow \mathbb{Z} \overset{3}\longrightarrow \mathbb{Z} ...
1
vote
1answer
80 views

Group completions and the induced homomorphisms

The group completion (aka Grothendieck group) of an abelian monoid $M$ is an abelian group $G(M)$ with a homomorphism $\iota:M \to G(M)$ of monoids satisfying the following universal property: for ...
1
vote
2answers
55 views

Cosets and Lagrange's Theorem

This question is under the topic of Cosets and lagrange theorem. Now Is it true that if $G$ is a group that contains a subgroup $H_1$ of order $n$ and a subgroup $H_2$ of order $k$, then $G$ must ...
2
votes
1answer
133 views

Finite abelian groups in which quotients of same order are isomorphic

Let $G$ be a finite abelian group which is isomorphic to direct sum of some elementary abelian groups and a cyclic group such that all summands have coprime orders. Are quotients of its all subgroups ...
1
vote
1answer
1k views

Invertability of Singular 2x2 Matrix with all same real values.

Question: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...
10
votes
4answers
655 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
1
vote
0answers
55 views

Product and quotient in Abelian groups

In linear algebra, for any vector space $V$ and its subspace $U$, there exists a subspace $W$ of $V$ such that $U\oplus W=V$ and $W\cong V/U$. Does similar property hold for Abelian groups? That is, ...
1
vote
1answer
86 views

Order of the factor group by the center equals order of the group

My question is: What are the implications if the order of the factor group $G/Z(G)$ is equal to the order of the group $G$? I know that $|G/Z(G)|=|G|$ if $|Z(G)|=1$ meaning the center is trivial. ...
0
votes
1answer
126 views

Free abelian subgroup of index 2.

Let $G$ be a group with the following presentation $G=gp(x,y \mid x^2=y^2=1)$. I need to know, what further information about $G$ can be derived from knowing that $G$ has a free abelian subgroup of ...
0
votes
2answers
292 views

All pairwise non-isomorphic abelian groups of order 67500?

$67500=2^2*3^3*5^4$ => $2*3*4=24$ pairwise non-isomorphic abelian groups. Is this correct?
3
votes
2answers
2k views

In an abelian group, the elements of finite order form a subgroup.

I need to show that elements of finite order in an abelian group form a subgroup of that group. Where do i start ?
1
vote
1answer
134 views

Free abelian groups.

Is the following implication true? If this is the case, how can this be shown? If $G$ is a one-relator (neither power nor commutator relation), two-generator group, then $G/G'$ (where $G'$ is ...
3
votes
1answer
160 views

Torsion group is a subgroup.

Let $G$ be an abelian group. Prove that $H =\{g \in G \;|\; |g| < \infty\}$ is a subgroup of $G$. Given an explicit example where this set is not a subgroup when $G$ is non-abelian. I am ...
2
votes
2answers
231 views

What is the difference between the words chord, tangent in (a) and (b)?

(a) If a function $g$ is continuous on the closed interval $[u,v]$, where $u<v$, and differentiable on the open interval $(u,v)$, then there exists a point $c$ in $(u,v)$ such that ...
2
votes
3answers
167 views

$G$ finite abelian, $\exists H < G : |G/H|$ is prime?

Let $G$ be a finite abelian group. Is there a subgroup $H < G $ s.t. the quotient $ G/H$ has prime order?
2
votes
2answers
110 views

Show that the mapping $x\rightarrow x^{-1}$ of $G$ onto $G$ is an isomorphism iff $G$ is abelian

Question : Show that the mapping $x\rightarrow x^{-1}$ of $G$ onto $G$ is an isomorphism iff $G$ is abelian, $x\in G $ I need some pointers for proving this .
6
votes
0answers
254 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
5
votes
0answers
136 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
1
vote
1answer
157 views

Power automorphism and abelian groups

First of all I'm not sure if "Power automorphism" is the correct term, so I apologize if it is not. "Let $G$ be an abelian group of order $n$, and $m$ an integer. $f:G\rightarrow G$ s.t. $f(a)=a^m$. ...
2
votes
0answers
49 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
3
votes
1answer
144 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
7
votes
2answers
1k views

Product of elements of a finite abelian group

Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$. Prove that if there is more than one element of order $2$ then $x=e$. What I've done so far: (#1 is just for ...
3
votes
1answer
594 views

Exponent of a finite abelian group

I have a very basic question: Let $G$ be a finite abelian group and let $m$ be the exponent of $G$. Then does there exist $g\in G$ s.t. o$(g)=m$ and if so, why? Many thanks in advance.
3
votes
1answer
211 views

Fundamental Theorem of Finite Abelian Groups and the mulitplicative group of a field.

(a) State the Fundamental Theorem of Finite Abelian Groups (invariant version). (b) Consider the group $K^{\times}$, under multiplication, of all non-zero elements in a field $K$. Let $A$ be a ...
0
votes
1answer
54 views

Proving that sum is invariant under bijections.

I am trying to prove the following: let $(G,+)$ be an abelian group, let $I_n^m = \{i \in \mathbb{N} : n \leq i \leq m\}$ and $I_r^k = \{i \in \mathbb{N} : r \leq i \leq k\}$. Let also $\varphi : ...
3
votes
1answer
122 views

A question on a subgroup of an abelian group

Let $X$ be an abelian group, $G$ a divisible subgroup of $X$ and $L$ a subgroup of $X$ such that $L\bigcap G\neq 0$. Let $nX=G$ for some $n$. For a positive integer $m$, define $H_{m}=\{x\in X;mx\in ...
2
votes
2answers
54 views

A question on finite index

Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $n(G/H)=0$ for some $n$. Can we derive that $H$ has finite index in $G$? ($nG$ denotes the subgroup $\{nx:x\in G\}$ for an arbitrary ...
4
votes
2answers
307 views

How to show this obvious and basic property of abelian groups?

I have a question that is probably very silly, but let's go. Let $(G,+)$ be an abelian group. In that case we know that $+$ is associative and commutative. This leads us to the following: if $\{a_i ...
2
votes
1answer
146 views

If $G$ is an abelian group and $G/H\cong G$, under what conditions does this imply $H=0$?

Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H\cong G$. What conditions on $G$ will force $H=0$?
12
votes
1answer
239 views

Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
6
votes
1answer
122 views

The structure of the group $(\mathbb{Z}/2^n\mathbb{Z})^*$

I really got stuck with this exercise, can you help me? This is the total exercise. Calculate $(1+4)^{2^{n-3}}\in (\mathbb{Z}/2^n\mathbb{Z})^*$, and show that the element $5$ has order $2^{n-2}$ ...
4
votes
2answers
671 views

Elementary divisors of an abelian group

From Advanced Modern Algebra (Rotman): Proposition 4.10 If $G$ is an abelian group and $p$ is prime, then $G/pG$ is a vector space over $\Bbb{F}_p.$ Definition If $p$ is prime and $G$ is a ...
1
vote
1answer
126 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
6
votes
3answers
100 views

Let $G$ be a finite p-primary abelian group. If a is an element of largest order in G, then $A= \langle a \rangle$ is a direct summand of G.

I was trying to read the proof from Advanced Modern Algebra (Rotman), but there was something that seemed confusing to me. It's only the last part that's confusing, but I put the whole proof anyway. ...
0
votes
0answers
58 views

A question on $n$-divisible groups

Let $G$ be an abelian group, and let $n^iG$ denote the subgroup $n^{i}G=\{n^{i}x;x\in G\}$. Is $\bigcap_{i=1}^{\infty}n^{i}G$, $n$-divisible?
2
votes
0answers
57 views

Definition of pure-essential extension

Let $B$ be a pure subgroup of an abelian group $A$. In his book "Infinite Abelian Groups", Academic Press, 2vols., Fuchs defines $A$ to be a pure-essential extension of $B$ if there is no nonzero ...
3
votes
0answers
87 views

Partition of Symmetric Group

For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...
3
votes
2answers
73 views

show that this proposed generalization is false.

Recall that lcm$(r,s) = rs$ if and only if gcd$(r,s) = 1$. Suppose you wish to generalize proposition 3 to: if $a$ has order $r$ and $b$ has order $s$, then $ab$ has order lcm$(r,s)$. show that this ...
1
vote
0answers
69 views

Finding subgroups to special groups

Let $G$ and $H$ be groups. Is there any possibility to find all (normal) subgroups of $G\times H$ and $G*H$? I really hope that this task is easier, if $G$ and $H$ are cyclic groups. I tried to find ...
0
votes
1answer
50 views

A question on $n$-divisible group for some $n$

Let $X$ be an abelian group and $G$ a divisible subgroup of $X$. Let $L$ be a subgroup of $X$ such that $nL\subseteq G$ for some $n$. Is $L/L[n]$ $n$-divisible? ($L[n]$ denotes the subgroup $\{x\in L; ...
4
votes
2answers
91 views

Reference Request for The Study of Abelian Groups

So I finished Lang's Algebra and after reading this partial Structure Theorem for abelian torsion groups that are not finitely generated , I've gotten interested in abelian groups, in particular ...
4
votes
3answers
101 views

Clarification regarding a group theory proof

In a group we have $abc = cba$. If $c \neq 1$, is the group abelian? See the following link. (I am new to this site but it is my understanding that you cannot PM authors, correct? Which is too bad ...
3
votes
1answer
57 views

A question on divisible groups

Let $p$ be a prime and $H=\prod_{n=1}^{\infty}\mathbb Z(p^{n})$ ($\mathbb Z(p^{n})$ is the finite cyclic group of order $p^{n}$). Is $H/t(H)$ divisible ($t(H)$ denotes the maximal torsion subgroup of ...
3
votes
1answer
79 views

Sum of two finite sets in torsion-free abelian groups

Suppose $G$ is a torsion-free abelian group (written additively) and $A$ and $B$ are two nonempty finite subsets (not subgroups) of $G$. Is it true that there is an element of $G$ which may be ...