Exercise · ZW-SYLOW-THEOREMS-005 · ★★☆ Intermediate
A group \(G\) of order \(20\) has a unique Sylow \(5\)-subgroup. How many elements of order \(5\) does \(G\) have?
Choose one option.
Show answer
- B. \(4\)
Explanation
That subgroup is cyclic of order \(5\), and all elements of order \(5\) lie in it, since they belong to some Sylow \(5\)-subgroup. Each of them differs from the identity: \(5-1=4\).
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-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: