find all sylow subgroups of D5 and D6SolutionLet n 3 be odd

find all sylow subgroups of D5 and D6

Solution

Let n >= 3 be odd. Then |Dn| = 2n, so every Sylow 2-subgroup of Dn has order
2 and has the form hgi = {1, g} where g 2 Dn is an element of order 2. Thus to
find all the Sylow 2-subgroups of Dn we need to find all elements of order 2 in Dn.

for D6 = {1, a, . . . a5, b, ba, . . . ba5} with |a| = 6, |b| = 2 and aba = b the
subgroupi has index 2 in D6 and is therefore a proper normal subgroup