# Math Help - GROUP theory

1. ## GROUP theory

Let Z(G) denote the center of a group G, then the order of a quotient group G/Z(G) cannot be
a) 4
b) 6
c) 15
d) 25

please send me the answer with solution.
bye

Let G be a finite group and Z(G) be a normal subgroup.
Then the quotient group Z(G) of G is ( |G| / |Z(G)| ).Here O(G) divides the O(Z(G)) .So we can take 15(factors 1,3,5,15) and 25(factors 1,5,25).SO the ans. would be (a) and ( b).

2. Originally Posted by cdniki
Let Z(G) denote the center of a group G, then the order of a quotient group G/Z(G) cannot be
a) 4
b) 6
c) 15
d) 25

please send me the answer with solution.
bye

Let G be a finite group and Z(G) be a normal subgroup.
Then the quotient group Z(G) of G is ( |G| / |Z(G)| ).Here O(G) divides the O(Z(G)) .So we can take 15(factors 1,3,5,15) and 25(factors 1,5,25).SO the ans. would be (a) and ( b).

The quotient group $G/Z(G)$ cannot be cyclic non-trivial, so the answer is...

Tonio