Are cyclic groups always abelian? [closed]

Solution 1:

Yes, a cyclic group is abelian. Here is why.

A cyclic group is generated by one generator, let's call this $g$. Now if $a = g^m$ and $b = g^n$ are two elements of the group, then $ab = g^{m}g^n = g^{n}g^m = ba$ (since $g$ commutes with itself).

Solution 2:

The following exercises (in order of increasing difficulty) will strengthen your understanding of the ideas presented above:

Exercise 1:

Give an example of an abelian group that is not cyclic.

Exercise 2:

Let $G$ be a group such that every proper subgroup of $G$ is cyclic. Is $G$ necessarily cyclic?

Exercise 3:

Let $G$ be a finite group with a unique maximal subgroup. Prove that $G$ is cyclic. (Hint: let $M$ be the unique maximal subgroup of $G$. Recall that the definition of "maximal subgroup" also requires that the subgroup be proper. Hence we can find $x\in G,x\not\in M$. Prove that $x$ generates $G$.)

Exercise 4:

Prove that a non-trivial abelian simple group is cyclic of prime order. (Recall that a group $G$ is said to be simple if it does not have any proper non-trivial normal subgroups.) (Hint: which subgroups of an abelian group are normal? What can you say about non-trivial groups with exactly two subgroups?)

Exercise 5:

Let $G$ be a finite group such that the quotient group $G/Z(G)$ is cyclic. Prove that $G$ is abelian, that is, prove that $G=Z(G)$. (Recall that $Z(G)$ is the center of $G$; namely, it is the set of all elements in $G$ that commute with every element of $G$.)

Challenging Exercises:

Exercise A:

Prove that an abelian group of order $6$ is cyclic. In fact, prove that there are exactly two isomorphism classes of groups of order $6$. Also, give representatives of these isomorphism classes.

Exercise B:

Prove that a group of order $p^2$ is abelian if $p$ is prime. (Hint: first prove (or use if you cannot prove) that if $G$ is such a group, then $Z(G)\neq 1$, that is, the center of $G$ is non-trivial. Then consider a case-by-case argument based on Exercise 5 above.)

Exercise C:

Prove that every group of order $35$ is cyclic. (Hint: first prove (or use if you cannot prove) that if $G$ is such a group, then $G$ has exactly one normal subgroup of order $5$ and of order $7$.)

Solution 3:

Yes, all cyclic groups are abelian. Here's a little more detail that helps make it explicit as to "why" all cyclic groups are abelian (i.e. commutative).

Let $G$ be a cyclic group and $g$ be a generator of $G$. Let $a,b \in G$. Since $g$ is a generator of $G$, all elements in $G$ can be expressed as integral powers of $g$ (or in the case the group operation on $G$ is additive, all the elements of $G$ can be expressed integral multiples of $g$). Then there exist $x,y \in \mathbb {Z}$ such that $a=g^x$ and $b=g^y$ (or $a = xg$ and $b = yg.$). Since $ab=g^xg^y=g^{x+y}=g^{y+x}=g^yg^x=ba$, (or $a+b = xg+ yg= yg + xg = b + a$) it follows that $G$ is abelian.

This proof helps show what Qia means in the first comment above by "an element always commutes with powers of itself." The key with cyclic groups is that all elements of a given cyclic group can be expressed in terms of one element in the group: as an integral power (or multiple) of the group's generator. (If a group $G$ has more than one generator, each element in $G$ can be expressed in terms of each of the group's generators).