Mark the following 6 statements by true or false with brief

Mark the following 6 statements by true or false with brief justification or discussion (i.e., giving a short proof or a counter example). The group Z_c is isomorphic to a subgroup of Z_10 Z_15 The group Z_10 Z_20 contains no element of order 8. The group Z_10 D_s is abelion. The ring Z_31 has no zero-divisors. The group U(16) is isomorphic to Z_1 Z_2

Solution

b) if (a,b) has a order 8 . b must have order 8 in respective group

c)Z_10 + D_3 is non-abelian since D_3 is not abelian.

d) 31 is a prime . So Z_31 is a field . has no zero divisors

e) U(16) = Z_4 + Z_2 false

U(16) = {1,3,5,7,9,11,13,15} is cyclic Z_4 + Z_2 is not