If a simple group G is of order 168 then can I find subgroup of order 7 of G ? If so, then what is the number of subgroups of G of order 7 ?
2026-03-27 10:31:41.1774607501
Can a Simple Group possess this property?
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Since $168=7\cdot 24$, Sylow's Theorem says that:
$n_7 \mid 24$
$n_7 \equiv 1 \bmod 7$
Since the group is simple, we have
Therefore, $n_7=8$ because $8$ is the only non-trivial divisor of $24$ that is congruent to $1 \bmod 7$.