Question

Question: If pis prime, prove that there are no simple groups of order 2p.

Short answer

It is proved that there are no simple groups of order 2p.

Step-by-step solution

Referring to Corollary 9.16 and Third Sylow Theorem

Corollary 9.16

Let G be a finite group and K is a Sylow p-subgroup for some prime p. Then, K is normal in G if and only if K is the only Sylow p-subgroup.

Third Sylow Theorem

The number of Sylow p-subgroups of finite group G divides$\left|G\right|$ and is of the form 1+pk for some non-negative integer k.

Given that p is prime.

Proving that there is no simple group of order 2p

Writing 2p as multiple of its factors as:

$2p=2.p$

ByThird Sylow Theorem, its p-subgroups should divide 2p and it should be in the form of 1+pk where,$k\ge 0$ .

Elements of 1+pk form are 1, 1+2p, 1+3p….

And divisors of 2p are 1, 2, p and 2p.

From the above, it can be seen that 1 is the only common number on both lists.

Therefore, 2p has exactly one 1-subgroup and this subgroup is normal byCorollary 9.16.

Consequently, no group of order 2p is simple.

Hence, it is proved that there is no simple group of order 2p.