5 votes

Best answer

Here we are given that the order of the group is 10 and a is the generator..

Hence a^{10} = e where e is the identity element but we dont know about a^{8}..Hence we have to write it in terms of a^{10} as that is a known result.

Let we have y = a^{40} which can be written as : a^{40} = (a^{10})^{4} = e^{4} = e ...(1)

Also , a^{40} = (a^{8})^{5} = e [ Follows from (1) ]

Hence it means that a^{8} is repeated 5 times in order to get the identity element e which is least number of times to do so.

Also from the corollary of the Lagranges' Theorem ,

Order of an element divides the order of the order of the group (The actual theorem is order of a subgroup divides the order of the group)

Here 5 divides the order of group i,e, 10..

**Hence order of a ^{8} = 5 **