Talk:Group with Order Less than 60 is Solvable

Is this proof enough?

Burnside's Theorem does not prove groups with order $30 = 2 \times 3 \times 5$ or $42 = 2 \times 3 \times 7$ are solvable. --Fake Proof (talk) 07:07, 28 July 2021 (UTC)

Heh! Good call. Looks in complete -- or that the person posting this proof up had a brainfart. Note "the smallest number with $3$ distinct prime factors is $2 \times 3 \times 5 = 60$" -- no it's not, $2 \times 3 \times 5 = 30$ as you correctly note.

One could approach this by eliminating all sphenic numbers less than $60$ -- oh good, that's just $30$ and $42$ as you note.

We have Group of Order 30 is not Simple and Group of Order 42 has Normal Subgroup of Order 7‎. Hence we can generate a composition series for them. Then Burnside's Theorem can be applied.

The source work says "See $59 \eta$" which is Simple Group of Order Less than 60 is Prime, which can then be used as a basis of a second proof, perhaps.

Feel free to work magic. --prime mover (talk) 07:40, 28 July 2021 (UTC)