# Abstract algebra question (math olympiad)

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1. Feb 16, 2012

Let G be a non-cyclic group of order pn where p is a prime number. Prove that G has at least p+3 subgroups.

Could anyone offer a solution to this problem?

2. Feb 16, 2012

### micromass

How many cyclic subgroups does such a group have?? (i.e. generated by one element).

3. Feb 16, 2012

### Deveno

have you tried induction, using n = 2 as your base case:

G = Zp x Zp

subgroups:

G
Zp x {0}
{0} x Zp
<(1,k)>, for p-1 choices of k (k ≠ 0)
{(0,0)}

total: 1+1+1+(p-1)+1 = p+3

for the induction step, you have two cases to prove:

G has a non-cyclic subgroup of order pk, k > 1 (easy case)
G has only cyclic subgroups (hard case)

for the second case (i haven't worked out all the details), i think you can use the fact that G has a non-trivial center, to reduce it to the case that G is abelian.

now if G is abelian and has only one subgroup of order pk, for every k < n, then G would be cyclic, and by supposition G is non-cyclic (appeal to sylow for existence of a subgroup of order pk)

so we can assume that for some 1< k < n, G has two distinct subgroups of order pk. but then we have two distinct subgroups of order p (by Cauchy), say H and K, and since G is abelian HK is a subgroup of G.

then: since G is supposed to be non-cyclic with only cyclic subgroups, HK must be cyclic, contradicting the fact that it has 2 distinct subgroups (H and K) of order p.

so G must not be abelian, and that should take care of it (unless i've done something stupid...could be).

4. Feb 16, 2012

### morphism

Here's a quick proof. Let $\Phi(G)$ be the Frattini subgroup of G (= intersection of maximal subgps of G). Then $G/\Phi(G) = (\mathbb Z/p\mathbb Z)^k$ for some $k\geq 1$ and moreover k=1 iff G is cyclic. Since G isn't cyclic, we see that $G/\Phi(G)$ is an elementary abelian p-group of rank at least 2, hence has at least (p^2-1)/(p-1)=p+1 subgroups of index p. Consequently G has at least p+1 subgroups of index p. Now throw in the trivial subgroup and G itself to see that G has at least p+3 subgroups.

5. Feb 16, 2012

### Deveno

by gosh, that's slick...let me ask you this, at what level do you think students get their first exposure to the Frattini subgroup?

6. Feb 16, 2012

### micromass

Sigh... I was hoping that the OP would discover the answer by our hints...

The different proofs are all nice though.

7. Feb 16, 2012

### Deveno

in a perfect world, which this is not, the OP will examine each reply and turn, and think deeply on them. if that happy event comes to pass, perhaps they will gain more than just "the answer".

8. Feb 16, 2012

### morphism

I'd say graduate-level, if at all.

re: micromass: in principle, I generally agree with you. That said, I think this problem is somewhat unlike the others that get posted here, and I genuinely think that it hasn't been ruined. Like Deveno says, each of the first three posts above contains ideas (and mini-exercises) for the OP to ponder.

9. Feb 19, 2012