Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$
3
votes
1answer
105 views
The cancellation property for finite abelian groups
I need some hints to prove that:
Let $A,B,C$ are finite abelian groups such that $A\oplus B\cong A\oplus C$. Prove that $B\cong C$.
I know that every finite abelian group can be written as a ...
-1
votes
1answer
190 views
Abelian Groups of order 2000
Classify, up to isomorphism, all abelian groups of order 2,000, giving the
standard form of each group in your list. (The standard form is also called
the invariant factor decomposition.)
1
vote
3answers
85 views
Suppose $G$ Abelian and $f:G\rightarrow \Bbb Z$ is surjective with kernel K. Show $G \cong H + K$ where $H \cong \Bbb Z$
Suppose $G$ abelian and $f:G\rightarrow \mathbb Z$ is surjective with kernel $K$.
Show that $G$ has a subgroup $H$ such that $H \cong \mathbb{Z}$
Show that $G \cong H\bigoplus K$
To get ...
0
votes
1answer
118 views
Group of homomorphisms
Let $A$ be a finite, Abelian, additive group. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$ denote the group of homomorphisms $f$ from $A$ to $\mathbb{Q}/\mathbb{Z}$. Take for granted that $A^{*}$ is an ...
0
votes
1answer
76 views
Number of subgroups of order p
Let p be a prime number and consider $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$. How many subgroups of order p does it have? Given any two subgroups $B_1, B_2 $ of ...
1
vote
1answer
73 views
Seeking a proof of: If any two Abelian groups of order $d$ are isomorphic, then $d$ is squarefree.
Suppose that $d \in \mathbb{N}$ satisfies the property that given any two Abelian groups $A_1, A_2$ of order $d$, $A_1 \cong A_2$. Prove that given any prime $p$, $p^2 \nmid d$. What can one say for ...
4
votes
1answer
226 views
Non-trivial homomorphism between multiplicative group of rationals and integers
Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
0
votes
1answer
63 views
showing G is abelian
If $G$ group of order $52$ includes a normal group of order $4$ then $G$ is abelian.
I did like this
$$
|G|=52=2^2\cdot 13
$$
let $H$ be normal group of order $4$.
$n_{13}=1$ thus $G$ has a $K$ ...
4
votes
2answers
164 views
How to show that these groups are not isomorphic?
Let $G$ is an abelian group and $tG$ is its torsion subgroup. If $p$ is a prime, how to show that:
$$t\bigg(\prod_{n=1}^{\infty}\mathbb Z_{p^n}\bigg)\ncong\sum_{n=1}^{\infty}\mathbb Z_{p^n}$$
I ...
2
votes
1answer
74 views
If every proper quotient is finite, then $G\cong\mathbb Z$
Here is my problem:
Let $G$ is an infinite abelian group. Prove that if every proper quotient is finite, then $G\cong\mathbb Z$.
And here is my incompleted approach:
I know that the quotient ...
2
votes
2answers
104 views
The ring of formal power series
Is there a simple proof/clarification of this statement?
The set of all formal power series in X with coefficients in a commutative ring R form another ring that is written R[[X]], and called the ...
19
votes
1answer
433 views
Recovering a finite group's structure from the order of its elements.
Suppose you know the following two things about a group $G$ with $n$ elements:
the order of each of the $n$ elements in $G$;
$G$ is uniquely determined by the orders in (1).
Question: How ...
0
votes
1answer
192 views
Pure Subgroups of $\mathbb Z\times\mathbb Z$
I am thinking of the following problem $^*$:
Given an example in which the subgroups generated by two pure subgroups ia not pure. (Hint: Look within a free abelian group of rank $2$.).
So, as ...
2
votes
2answers
67 views
Bases for $\Bbb Z^n$ containing a given vector
I am trying to prove the following theorem on finitely generated free abelian groups (which thus for simplicity may be assumed to be $\Bbb Z^n$):
Let $\alpha \in \Bbb Z^n$ be such that for all $k ...
3
votes
1answer
102 views
What do you call groups where you can apply the group operation a fractional number of times?
I have a group $G$, and for all $g \in G$ and $a,b \in \mathbb{Z}$ it makes sense to talk about the element $g^{a \over b} \in G$.
To get some intuition, I've been thinking about what it would mean ...
0
votes
3answers
238 views
G is an abelian group. Prove $G^{(n)}$ is a subgroup of G
Let G be an abelian group. Prove that
$$G^{(n)} = \{g \in G | g^n = 1_G \}$$
is a subgroup of G.
How do I go about doing this?
I understand that $G^{(n)}$ is basically the set of all elements ...
1
vote
2answers
54 views
Commutative Groups and Quotients
Let (A,+) and (B,+) be commutative groups and suppose that A is isomorphic to B. Prove that $A/dA$ is isomorphic to $B/dB$, where $d \in \mathbb{N}$.
Any thoughts on this one?
4
votes
2answers
145 views
Frattini subgroup of additive group of rational numbers
Show that Frattini subgroup of additive group of rational numbers $(\mathbb{Q},+)$ is itself, or
$$\Phi(\mathbb{Q})=\mathbb{Q}$$
PS. My strategy is prove that group $(\mathbb{Q},+)$ hasn't maximal ...
3
votes
2answers
90 views
Direct limit of $\mathbb{Z}$ homomorpisms
What is the direct limit of the following sequence of $\mathbb{Z}$ homomorphisms (as groups)? $$ \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{3} \mathbb{Z}\xrightarrow{5} ...
1
vote
1answer
58 views
$G/S$ is torsion-free?
There is a well-known theorem that:
If $S\le G$ and $\frac{G}{S}$ is torsion-free, so $S$ is pure in $G$.
Please hint me about the reverse. If $S$ is pure in $G$, then will $\frac{G}{S}$ is ...
5
votes
0answers
40 views
Shift operator on locally compact groups
Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively).
...
2
votes
1answer
91 views
p-group: cyclic $n \leq 1$ | abelian $n \leq 2$
$p$ prime number, $n$ a non-negative integer, $G$ group.
(a) $\forall G$ with $|G| = p^{n}$ cyclic $\Leftrightarrow$ $n \leq 1$.
(b) $\forall G$ with $|G| = p^{n}$ abelian $\Leftrightarrow$ $n \leq ...
1
vote
2answers
52 views
inverse of even number of elements in a group
if an abelian group with |G|=n where n is odd. if i take out the identity i'm left with even # of distinct elements. can this mean that each element has an inverse which is not itself??
not a homework ...
0
votes
1answer
63 views
Group homomorphisms between two abelian groups with different kernel
Does there exist two abelian groups $A,B$ with an epimorphism $f: A\to B$, and two other abelian groups $A', B'$ along with an epimorphism $g: A'\to B'$ such that $A\cong A'$, $B\cong B'$ and $ker\,f ...
6
votes
1answer
131 views
$G$ is isomorphic to a subgroup of $H$ and vice versa
Let $G$ and $H$ are two divisible groups that each of which is is isomorphic to a subgroup of the other, then $G\cong H$.
What I've done is to use the injective property for both groups:
...
4
votes
0answers
121 views
Direct sum of Prüfer groups and $\mathbb Q/\mathbb Z$
It can be easily shown that, the Prüfer $p$-group $\mathbb Z(p^\infty)$ is isomorphic to multiplicative group $$R_p=\{e^{2\pi ik/p^n}|k\in\mathbb Z,n\geq0\}$$ Now I want to prove that:
...
1
vote
2answers
60 views
$G$ is a reduced Group
Here is the problem:
Let $G=\langle x_0,x_1,x_2,\ldots\ |px_0=0,x_0=p^nx_n, \text{all } n\geq1\rangle$. Prove that $G/\langle x_0\rangle$ is a direct sum of cyclic groups and is reduced.
The ...
4
votes
2answers
168 views
Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?
I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
1
vote
0answers
64 views
If $G=\prod_{p\in P}\mathbb Z_p$ then $\frac{G}{tG}$ is divisible.
I want to show that:
If $G=\prod_{p\in P}\mathbb Z_p$, wherein $P$ is the set of all primes, then $\frac{G}{tG}$ is divisible.
I know that $tG$ is not a direct summand and if $x\in G$ wants to ...
1
vote
1answer
70 views
$F/H$ has an element of infinite order?
Here is a problem:
If $F$ is a free abelian group of rank $n$ and $H$ is a subgroup of rank $k<n$, then $F/H$ has an element of infinite order.
What I did:
I assume $F=\langle ...
2
votes
1answer
74 views
$\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$
I want to prove that:
If $F$ is a free abelian group of rank $n$, then $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$ with ...
2
votes
1answer
78 views
Verifing some properties about $G$
I have the following problem$^*$:
Prove that the group $G$ having generators and relations respectively $$X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$$ is an infinite ...
1
vote
1answer
43 views
$G$ is torsion-free group then $G/\langle X\rangle$ is torsion
Honestly, I have been thinking on this problem for hours but couldn't find a way:
Let $G$ is torsion-free group and $X$ is a maximal independent subset, then $G/\langle X\rangle$ is torsion.
I ...
1
vote
1answer
68 views
Divisible abelian $q$-group of finite rank
What does "finite rank" mean in the context of divisible abelian $q$-group?
A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
1
vote
1answer
54 views
Exact sequence of abelian groups, property transfers
We had the statement that with an exact sequence of multiplicatively written abelian groups
$U \mapsto V \mapsto W \mapsto X \mapsto Y$ and in $U$, $V$, $X$, $Y$ every group element has a unique ...
2
votes
2answers
104 views
Prove $H \times G$ is commutative iff $H, G$ are commutative
This is another proof question I am asking about, can someone give me tips on how to answer these questions?
My question says:
"Let $H,G$ be arbitrary groups. Prove that $H \times G$ is commutative ...
1
vote
2answers
161 views
determining number of subgroups
If $G$ is an abelian group of order 72, do we know how many subgroups of order 8 it has?
Just because it's a divisor doesn't mean that there is a subgroup of that size. But I'm wrong. Why?
0
votes
1answer
143 views
Groups of the same order that are nonisomorphic
I'm reading A First Course in Abstract Algebra by Fraleigh and I've reached a point where I feel like I'm supposed to have understood something more from the chapter than what is actually stated. I've ...
4
votes
1answer
125 views
Smallest pure subgroup containing a fixed subgroup
I will ask a slightly more precise question then in the title.
Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
3answers
166 views
Splitting exact sequences of finite abelian groups
I would like to find a condition for an exact sequence of abelian groups
$$
0\to H\to G\to K\to 0
$$
to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
3
votes
1answer
148 views
An indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable
I'd like to find an indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable, and I'm running out of ideas:
The only indecomposable $\mathbf{Z}$-modules I know are $\mathbf{Z}$, ...
3
votes
3answers
99 views
Constructing a basis for finite abelian groups
Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$.
I would like to ...
0
votes
0answers
37 views
When can we preserve injectivity of $\mathbb{Z}$-modules after tensoring by $\mathbb{Z}/p\mathbb{Z}$
This question can be asked in a more general setting (the one from the title, for instance), but I am interested in the algebro-geometric case :
Let $A$ be an abelian variety over an algebraically ...
0
votes
2answers
51 views
Elementary Question about Torsion Subgroups
Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$.
Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime.
Denote by $G[a]$ resp. $G[b]$ the $a$-resp. ...
3
votes
1answer
125 views
When abelian group is divisible?
By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$.
Let $A$ be abelian group with no proper subgroups of finite index. How can I prove ...
3
votes
1answer
139 views
Linear algebra of finite abelian groups
Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and
let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
5
votes
1answer
134 views
Adjoint of forgetful functor between category of vector spaces and category of abelian groups
I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group.
My question is, is ...
0
votes
0answers
54 views
Abelian group problem with three consecutive intergers [duplicate]
Possible Duplicate:
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
If $G$ is a group such that $ (a \circ b)^i = a^i \circ b^i $ for ...
4
votes
2answers
177 views
Subgroups of $\Bbb{R}^n$ that are closed and discrete
I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$.
Base ...
5
votes
6answers
1k views
Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.
I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
