There is a claim saying that if both $G'/G''$ and $G''$ are cyclic groups, then $G''=1$, where $G'$ is the derived subgroup of the group $G$. I have been thinking of this by focusing the N/C Lemma to clear the problem for myself. I need a useful igniting hint(s). Furthermore, may I ask: are these kinds of groups well known? Of course, any group satisfying the above conditions will be metabelian and obviously is soluble.
When $G'$/$G''$ and $G''$ both are cyclic groups
8
$\begingroup$
abstract-algebra
group-theory
-
1If $G'=\mathbb{Z}/4$ and $G''=\mathbb{Z}/2$ then $G'/G''=\mathbb{Z}/2$. – 2011-05-25
-
0Or are $G',G''$ supposed to be subgroups of a larger group? – 2011-05-25
-
0@Joe: your description is really right. I don't think G′,G′′ need to be subgroups of any larger one. – 2011-05-25
-
0$G''=Z_p$, $G'=Z_p\times Z_q$, $G'/G'' = Z_q$. ;-) – 2011-05-25
-
0Are you sure $G'$ and $G''$ are not supposed to be $[G,G]$ and $[G',G']$? Otherwise, it is clearly not true. – 2011-05-25
-
0What is $G'$ and what is a calim? – 2011-05-25
-
0@Tobias: @Qiaochu: You are right. G′ and G′′ are [G,G] and [G′,G'] reletively. – 2011-05-25
-
1@Basil R: you should edit your title and question. – 2011-05-25
-
1$G''=1$ is the same thing as saying $G'$ is abelian, so you're trying to prove that if $G'/G''$ and $G''$ are both cyclic then $G'$ is abelian. – 2011-05-25
-
0@Gerry: You said what I exactly meant. :) – 2011-05-25
1 Answers
6
This is theorem 9.4.2, page 146, in M. Hall's textbook on the Theory of Groups.
You are going in the right direction. It uses the N/C theorem, as in, the normalizer modulo the centralizer is a subgroup of the automorphism group.
Another hint: It is very similar to showing "If G/Z(G) is cyclic, then G is abelian.".
These groups were known as "metacyclic groups" for a few decades, though the name is now used slightly differently.
A special case where G′ and G/G′ have coprime order is very special: these are exactly the groups in which all Sylows are cyclic. They are also known as "Z-groups", though again the name means different things to different people.