Homework SourseProve that a group of order 10 is either isomorphic to Z10 o
Prove that a group of order 10 is either isomorphic to Z(1)0 or the dihedral group D5. (Hint: Cauchy theorem)
Prove that a group of order 10 is either isomorphic to Z(1)0 or the dihedral group D5. (Hint: Cauchy theorem)
Prove that a group of order 10 is either isomorphic to Z(1)0 or the dihedral group D5. (Hint: Cauchy theorem)
Solution
The two groups are isomorphic: indeed let us consider the subgroup
K of D10 generated by x2 and y. The element x2 has clearly order 5, moreover
x2y = yx-2 = y(x2)-1. This implies that K is isomorphic to D5. Moreover K
is normal since K has index 2 in D10. Since the intersection H \\K is trivial,
D10 is isomorphic to D5 H.
