-1
votes
3answers
43 views

$G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
0
votes
0answers
58 views

Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
0
votes
0answers
14 views

Cyclic Factor group abelian proof [duplicate]

Show that if G is nonabelian, then the factor group G/Z(G) is not cyclic. I started to prove this via contrapositive. If G/Z(G) is cyclic, then G is abelian. I'm messing around with elements and ...
1
vote
1answer
34 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
2
votes
2answers
90 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
1
vote
1answer
25 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
0
votes
1answer
47 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
2
votes
2answers
142 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
1
vote
1answer
59 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
-3
votes
3answers
93 views

Let $G = \{1, a, b, c\}$ be a group of order 4…Exist two groups of order $4$.

Let $ G = \{1, a, b, c\}$ be a group of order 4. Show that, if $G$ is cyclic $G \cong \mathbb Z_4,$ and if $G$ is not cyclic then $G \cong K_4.$ It now follows that there are only 2 groups of order ...
4
votes
0answers
114 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
7
votes
1answer
150 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
1
vote
2answers
83 views

How do we prove this fact about cyclic groups?

Prove that an Abelian group of order 33 is cyclic. Can we take an element a of order 3 and an element b of order 11 and say, |ab|=33?
0
votes
3answers
152 views

Prove or disprove that there is an abelian, noncyclic group of order 52.

So I've heard one must invoke Sylow's theorems in order to break down something like this. So far I know that there is a subgroup of order 13 in G, and that it's the only subgroup of order 13 in G. To ...
0
votes
0answers
70 views

Showing that $ U(2^n) $ is not a cyclic group for $ n \geq 3 $ [duplicate]

Could anyone please explain to me why $ U(2^n) $ is not a cyclic group for $ n \geq 3 $? I need help on this because I have an algebra exam tomorrow. Thanks!
4
votes
1answer
183 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 ...
2
votes
2answers
93 views

Finding a subgroup in the Center with order 91

Question: Let G be a group of order $455=5\cdot 7\cdot 13$. Show that exists a normal subgroup $ H<G: |H|=91$ and $H\subseteq Z(G)$. Show that G is an Abelian and cyclic group. Solution: So ...
0
votes
3answers
523 views

Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian [duplicate]

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
2answers
362 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 ...
5
votes
2answers
195 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 ...
3
votes
5answers
226 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 ...
6
votes
3answers
399 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 ...
2
votes
3answers
137 views

Understanding a Theorem regarding Order of elements in a cyclic group

This is part of practice midterm that I have been given (our prof doesn't post any solutions to it) I'd like to know whats right before I write the midterm on Monday this was actually a 4 part ...
1
vote
3answers
619 views

Show that all abelian groups of order 21 and 35 are cyclic.

Show that all abelian groups of order 21 and 35 are cyclic. I have no idea on how to start. Can anyone give some hints?
8
votes
5answers
650 views

Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
2
votes
1answer
124 views

$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$

Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
6
votes
6answers
3k 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.
5
votes
1answer
444 views

Group extensions of cyclic groups

Let $A$ be an infinite cyclic group and $B$ be a cyclic group of order $n$. Suppose $$0 \to A \to G \to B \to 0$$ is a short exact sequence of abelian groups. What could $G$ be? It is clear enough ...
3
votes
3answers
252 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
9
votes
5answers
2k views

Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!