Question

Question: Prove that there are no simple groups of order 30.

Short answer

It is proved that there is no simple group of order 30.

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.

Proving that there is no simple group of order 30

Writing 30 as multiples of its factors as:

$30=2.3.5$

Consider Sylow 5-subgroups.

ByThird Sylow Theorem, its 5-subgroups should divide 30 and it should be in the form of 1+5k, where$k\ge 0$ .

Elements of 1+5k form are 1, 6, 11, 16, 21, 26, and 31.

And divisors of 30 are 1, 3, 5, 15, and 30.

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

Therefore, 30 has exactly one 5-subgroup. This subgroup is normal byCorollary 9.16.

Consequently, no group of order 30 is simple.

Hence, it is proved that there are no simple groups of order 30.