Exercise 2154Algebra › GroupsM1Added 3 Nov 2026

Sylow's third theorem

#Sylow#congruence#number of subgroups

Statement

State Sylow's third theorem and compute the possible numbers of Sylow $3$ -subgroups in a group of order $45$ .

Hint

$|G|=45=3^2\cdot5$ . Let $n_3$ be the number of Sylow $3$ -subgroups. Then $n_3\equiv1\pmod 3$ and $n_3\mid5$ .

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Statement. The number $n_p$ of Sylow $p$ -subgroups satisfies:
- $n_p\equiv1\pmod p$ ;
- $n_p\mid m=|G|/p^a$ .

Application. $|G|=45=3^2\cdot5$ . $n_3\equiv1\pmod3$ and $n_3\mid5$ : possible values are $1$ and $5$ .

Conclusion. $n_3\in\{1,5\}$ . If $n_3=1$ , the Sylow $3$ -subgroup is normal.