E Order of Elements in Quotient Groups E Order of Elements i

E. Order of Elements in Quotient Groups E Order of Elements in Quotient Groups Let G be a group, and H a normal subgroup of G. Prove the following:

Solution

For the quotient group G/H we know that its elements are of the form Hg, where g G. Also, He = H is the identity of G/H. The multiplication of elements Hg and Hg₁ is given by (Hg)(Hg1 ) = Hgg₁. Now if m is the order of an arbitrary element a of (G:H) , then, (Ha)^m =Ha^m = H. This means that the order of Ha is a divisor of m. Thus, the order of every element of G|H is a divisor of m.