Exercise · ZW-SYLOW-THEOREMS-009 · ★★★ Advanced
If \(P\) is a Sylow \(3\)-subgroup of \(G\) and \(g\in G\), the conjugate \(gPg^{-1}\) is:
Choose one option.
Show answer
- C. another Sylow \(3\)-subgroup, of the same order as \(P\)
Explanation
Conjugation is an isomorphism, so \(gPg^{-1}\) has the same order as \(P\) and is also a Sylow \(3\)-subgroup. It equals \(P\) only when \(g\) normalizes it, not always.
Review the concept: Sylow theorems
A finite group \(G\) of order \(p^k m\), with \(p\) prime and \(p\nmid m\), has subgroups of order \(p^k\), all conjugate to each other; their number \(n_p\) is restricted.
\[n_p\equiv1\pmod p,\qquad n_p\mid m\]
More practice problems: sylow theorems
- ZW-SYLOW-THEOREMS-001 How many Sylow 5-subgroups can a group of order 15 have?
- ZW-SYLOW-THEOREMS-002 In a group of order 12, which is a possible value of n_3?
- ZW-SYLOW-THEOREMS-003 What is the order of a Sylow 2-subgroup of a group of order 72?
- ZW-SYLOW-THEOREMS-004 A group of order 56 has n_7 equal to:
- ZW-SYLOW-THEOREMS-005 A group G of order 20 has a unique Sylow 5-subgroup. How many elements of order 5 does G have?
- ZW-SYLOW-THEOREMS-006 A group of order 21 has n_7=1. What follows?
- ZW-SYLOW-THEOREMS-007 How many Sylow 2-subgroups does the symmetric group S_3 have?
- ZW-SYLOW-THEOREMS-008 Every group of order 35 is: