Exercise · ZW-SYLOW-THEOREMS-003 · ★☆☆ Basic
What is the order of a Sylow \(2\)-subgroup of a group of order \(72\)?
Choose one option.
Show answer
- B. \(8\)
Explanation
Since \(72=2^3\cdot3^2\), a Sylow \(2\)-subgroup has the largest power of \(2\) dividing \(72\), namely \(8\). The number \(9\) would be the order of a Sylow \(3\)-subgroup.
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-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:
- ZW-SYLOW-THEOREMS-009 If P is a Sylow 3-subgroup of G and g∈ G, the conjugate gPg^-1 is: