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in Set Theory & Algebra 111 views

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

Hence  a10  =  e  where e is the identity element but we dont know about a8..Hence we have to write it in terms of a10 as that is a known result.

Let we have   y  =  a40   which can be written as :    a40  =   (a10)4   =  e4   =   e    ...(1) 

                                                            Also ,        a40   =  (a8)5     =  e     [ Follows from (1) ]

Hence it means that  a8 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  a8   =   5 

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