Solve Induction Problem: Show Group G of Order p^n is Cyclic

[tyrannosaurus]

Homework Statement

Suppose the G is a group of order p^n, where p is prime, and G has exactly one subgroup for each divisor of p^n. Show that G is cyclic.

Homework Equations

The Attempt at a Solution

First, the inductive basis holds for n=1. Now for each k>=1, we assume that the claim is true for a group of order p^k where k < n. We shall show that the claim is also true for a group of order p^n.

Note that in a general case, the group of order p^n is not necessarily abelian. However, by the given inductive hypotheis, we show that the above group G of order p^n is abelian. We know that Z(G) is non-trivial for every nontrivial finite p-group and is a subgroup of G.

From this I am suppose to use the G/Z(G) theroem to show that G/Z(G) is cyclic and thus that G is abelian. However, I don't see how I this is true, for example what if |G|=p^3 and |Z(G)|=p, then G/Z(G)= p^2, which does not mean that G/Z(G) is cyclic. (If I remember right G/Z(G) has to be trivial cyclic)
From this I can use the Fundamental Theorem of Finite Abelian Groups to get the G is isomorphic to Zp+Z(p^n-1) or Zp and then conclude that it is isomorphic to Zp.

 

[Dickfore]

What are the divisors of p^{n}?

 

[tyrannosaurus]

1, p, p to m where m divides n and p^n. We know that we can't have 1 because Z(G) needs more than one lement

 

[Dickfore]

Every group has 2 subgroups, namely \{e\} and G. So, there is always a subgroup with 1 element.

 

[tyrannosaurus]

{e} is always a subgroup, but in are case Z(G) cannot equal e since are group G is of order p^n, where p is a prime.

 

[Dickfore]

I'm sorry. Maybe it's trivial, but I am unfamiliar with the notation Z(G). What do you mean by it?

 

[tyrannosaurus]

Z(G) is the center of the group G, for a to be an element of Z(G) a has to commute with all elements of G. Z(G) can be trivial (i.e just e) in some cases, but in this one it can't since |G|=p^n where p is a prime (p0groups have non-trivial centers).

 

[Office_Shredder]

You're probably supposed to use the inductive hypothesis on the quotient G/Z(G)