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Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $336$; this self-answered question aims to fill that gap. (See Prove there are no simple groups of even order $<500$ except orders $2$, $60$, $168$, and $360$., the sole answer of which describes an outline for proving the titular statement but does not describe how to establish the claim for order $336$.) Other answers are, of course, welcome.

How does one show that there are no simple groups of order $336$?

This question appears in Dummit & Foote's Abstract Algebra as Exercise §6.2.9.

Travis Willse
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  • N.b. This question asking the same thing but was closed as a duplicate: https://math.stackexchange.com/questions/4710264/group-of-order-336-cannot-be-simple But the question of which it is claimed a duplicate is exactly the question https://math.stackexchange.com/questions/3433417/prove-there-are-no-simple-groups-of-even-order-500-except-orders-2-60/3433540 mentioned in the above question statement---which categorically does not resolve the case of this particular order. – Travis Willse Jun 22 '23 at 03:31

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Suppose $G$ were simple of order $336 = 2^4 \cdot 3 \cdot 7$. Sylow's Theorems imply that $n_7 = 1 \pmod 7$ and $n_7 \mid 264$; simplicity imposes $n_7 > 1$ leaving $n_7 = 8$ as the only possibility. So, we can identify $G$ with a subgroup of $S_8$, and since $G$ is simple, it contains no subgroup of index $2$, hence $G \leq A_8$. Now, let $P$ be a Sylow-$7$ subgroup; Frattini's Argument gives that $$|N_{A_8}(P)| = \frac{1}{2} |N_{S_8}(P)| = \frac{1}{2} (7)(7 - 1) = 21 .$$ But $|N_G(P)| = \frac{|G|}{n_7} = \frac{336}{8} = 42$, and $42 \not\mid 21$, which contradicts the fact that $N_G(P) \leq N_{A_8}(P)$.

Remark Cf. this answer, which uses a similar technique to resolve the case of order $264$.

A related but distinct approach is given in $\S$3 the cited article (as far as I know, the latter is the earliest proof of the claim for this order).

F.N. Cole, "Simple Groups from Order $201$ to Order $500$," Amer. J. Math. 14(4) (October 1892), pp. 378$-$388.

Travis Willse
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