Chapter 36

A Sylow 3-subgroup of a group of order 12 has order.

Through 4 , fill in the blanks. A Sylow 3-subgroup of a group of order 12 has order _______.

A Sylow 3 -subgroup of a group of order 54 has order.

Through 4 , fill in the blanks. A sylow 3 -subgroup of a group of order 54 has order ______.

Find all Sylow 3 -subgroups of \(S_{4}\) and demonstrate that they are all conjugate.

Find two Sylow 2-subgroups of \(S_{4}\) and show that they are corjugate.

Let \(H\) be a subgroup of a group \(G\). Show that \(G_{H}=\left\\{g \in G \mid g H g^{-1}=H\right\\}\) is a subgroup of \(G\).

Let \(G\) be a finite group and let primes \(p\) and \(q \neq p\) divide \(|G|\). Prove that if \(G\) has precisely one proper Sylow \(p\)-subgroup, it is a normal subgroup, so \(G\) is not simple.

Show that every group of order 45 has a normal subgroup of order \(9 .\)

Let \(G\) be a finite group and let a prime \(p\) divide \(|G|\). Let \(P\) be a Sylow \(p\)-subgroup of \(G\) and let \(H\) be any \(p\)-subgroup of \(G\). Show there exists \(g \in G\) such that \(g H g^{-1} \leq P\).

Show that every group of order \((35)^{3}\) has a normal subgroup of order 125 .

Show that every group of onder \((35)^{3}\) has a normal subgroup of order \(125 .\)

Show that there are no simple groups of order \(255=(3)(5)(17)\).

Show that there are no simple groups of order \(p^{\prime} m\), where \(p\) is a prime, \(r\) is a positive integer, and \(m

Let \(G\) be a finite group and let \(P\) be a normal \(p\)-subgroup of \(G .\) Show that \(P\) is contained in every Sylow \(p\)-subgroup of \(G\).