Give an example of a non-Abelian group that has exactly four elements of order 10. Solution D10 is isomorphic to the semidirect product Z5?Z2. Interestingly, the unicode description of ? is \"right normal factor semidirect product\", even though all the references I have seen that use the symbol use it to refer to the factor on the _left_ being the normal subgroup. the order of fr in D10 is 2, not 10: (fr)2 = frfr = ffr-1r = e. Indeed, D10 contains no elements of order 10. The easiest way to do this is to note that Z10 under addition contains exactly four elements of order 10 - namely, 1, 3, 7, and 9. Of course, Z10 is abelian, but you know that D20 contains a normal subgroup isomorphic to Z10, and is not abelian. Thus, in D20, r, r3, r7, and r9 are elements of order 10, and it is easy to verify that D20 has no other elements of order 10. So that would be the simplest example..

1. Give an example of a non-Abelian group that has exactly four elements of order 10.
Solution
D10 is isomorphic to the semidirect product Z5?Z2.
Interestingly, the unicode description of ? is "right normal factor semidirect product", even
though all the references I have seen that use the symbol use it to refer to the factor on the _left_
being the normal subgroup.
the order of fr in D10 is 2, not 10: (fr)2 = frfr = ffr-1r = e. Indeed, D10 contains no elements of
order 10.
The easiest way to do this is to note that Z10 under addition contains exactly four elements of
order 10 - namely, 1, 3, 7, and 9. Of course, Z10 is abelian, but you know that D20 contains a
normal subgroup isomorphic to Z10, and is not abelian. Thus, in D20, r, r3, r7, and r9 are
elements of order 10, and it is easy to verify that D20 has no other elements of order 10. So that
would be the simplest example.