Questions tagged [cyclic-groups]

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

A group $G$ is cyclic if it can be generated by a single element $a$. This means that any element of a cyclic group has the form $a^n$ for some integer $n$. Notationally, we often write that $G$ is isomorphic to $\langle a \rangle$. Since

$$a^na^m=a^{n+m}=a^{m+n}=a^ma^n\,,$$

cyclic groups must be abelian. Note though that the generator is not necessarily unique: for example the cyclic group $\mathbf{Z}/7\mathbf{Z}$, consisting of the elements $\{0,1,\dotsc,6\}$ and equipped with the operation of addition modulo $7$, can be generated by any of its non-identity elements.

Cyclic groups are completely classified. Up to isomorphism, $\mathbf{Z}$ equipped with addition is the only infinite cyclic group. Every finite cyclic group is isomorphic to a group of the form $\mathbf{Z}/n\mathbf{Z}$, a quotient of the integers under addition modulo $n$.

Cyclic groups are incredibly useful in describing the structure of finite abelian groups. By the classification theorem of finite abelian groups, every finite abelian group is isomorphic to a direct sum of cyclic groups, each having order a power of a prime.

2243 questions
7
votes
1 answer

When p groups are cyclic?

Let $|G|=p^n$ and have only one subgroup of order $p^{(n-1)}$.Then G is cyclic.I am trying it in many ways bt get nothing. What I get :The unique subgroup is normal in $G$, Center meets the subgroup non trivially, No element…
Via
  • 425
6
votes
1 answer

Why $c(a_1 \ a_2 \dots \ a_k)c^{-1}=(c(a_1) c(a_2)… c(a_k))$?

We investigate on an arbitrary $a_i$ : $c(a_1 \ a_2 \dots \ a_k)c^{-1}(a_i)$. First step, $c(a_i)=a_k$. Second step, $(a_1 \ a_2 \dots \ a_k)(a_k)=a_{k+1}$, Third step, $c^{−1}(a_{k+1})=? $. Any answer that I read in MSE was not helpful to…
user200918
4
votes
4 answers

Proof: Every Cyclic Group is Abelian

Dr. Pinter's "A Book of Abstract Algebra"'s chapter on Cyclic Groups presents the exercise: Prove that every cyclic group is abelian. Here's my attempt: By Theorem 1 (of this chapter): (i): For every positive integer $n$, every cyclic …
3
votes
2 answers

Product of Elements in a Finite Cyclic Group with Odd or Even Order

Assuming that G is a finite cyclic group, let "a" be the product of all the elements in the group. i. If G has odd order, then a=e. Is this because there are an even number of non-trivial elements must have their inverses within the non-trivial…
Luke8ball
  • 323
3
votes
1 answer

Formal definition of cyclic order

I want to say the nodes $0,1,2 \ldots n-1$ are in cyclic order and want to express that in terms of mathematical notation. Say $0 \prec_c 1$, $1 \prec_c 2$, $n-1 \prec_c 0$ etc. What will be the formal definition of cyclic ordering? How do I define…
max
  • 281
2
votes
1 answer

$\rm{Aut}(G)$ is not cyclic when $G$ is not abelian

we have a $G$ which is not abelian and we need to prove that the group of all automorphisms is not cyclic. any ideas?
idan
  • 21
2
votes
1 answer

How do I prove that the order of an element is p?

I have the following problem: Let G be a group of order $p^2$ and assume that G is not abelian. I have just shown that $Z(G)$ is cyclic and has order p. Now I want to show that for each element $g\in G\setminus Z(G)$ we have…
user123234
  • 2,885
2
votes
2 answers

The order of a group versus its cyclic subgroup

I read a statement in my textbook, which was taken as a premise, that for some element $g$ of a finite group $G$, that the order of the element $g$ is the same as the cyclic subgroup, $$. I think I might be confusing the order of an element with…
user465188
1
vote
0 answers

Cyclic $\mathbb{Z}_{2p^k}^*$ group

I'm new in group theory and have some problems. I have got difficulties with understanding my notes from lecture. Maybe you can help me? I have this proof that $\mathbb{Z}_{2p^k}^*$ ($p>2$ prime) is a cyclic group. It uses the fact that …
xan
  • 2,053
1
vote
1 answer

Question on Cyclic Subgroups

Suppose we have a finite group, $G$ with some element, $g$. Is it plausible to say that, $\forall g \in G$, there exists a cyclic subgroup, $\langle g\rangle$, of $G$? I know, by definition, what $\langle g\rangle$ amounts to and that it's order is…
user465188
1
vote
2 answers

Direct sum of cyclic groups

Is a direct sum of cyclic groups cyclic? I know every abelian group is a direct sum of cyclic groups of prime power orders, but I can't make use of this.
user390960
1
vote
2 answers

What is the cyclic subgroup of integers(Z) generated by -1 under +?

I am trying to understand cyclic, and need to know what the cyclic subgroup of integers (Z) generated by -1 under + ?
m.b
  • 11
1
vote
2 answers

Let G be a non trivial group with no non trivial proper subgroup. Prove that G is a group of prime order.

This question is from cyclic group exercise. I have read the theorems of cyclic groups. But I could mot answer with proper language. Help me to solve.
user409382
  • 71
  • 7
1
vote
2 answers

How to show $2$ and $3$ are generators of the additive group $Z/(5)$.

This may sound like an easy question, but I never learned cyclic groups due to my time constraint of my class and that the professor ran out of time to teach it. However, he left me with a problem to solve. The question is "Show that both $2$ and…
1
vote
1 answer

Generators in cyclic group - specific example

I need to show that $(\mathbb{Z}_4 \ (0), \times)$ with multiplication modulo 5 has 2 generators; $2$ and $4$. I understand why $2$ is a generator, but I don't get why $4$ is. Edit: I'm not sure that's formatting correctly but it should be the…
Alice
  • 185
1
2 3