Question

the number of element of order 5 in the group z25+z5 is

Answer 1

Elements of order 5 in the group Z5 ⊕ Z25 can be expressed as:

lcm (|x|, |y|) = 5

Here, x belongs to Z5 and y belongs to Z25

Number of elements of order 5, when |x| = 1 and |y| = 5 are: φ(1).φ(5) = (1).(4) = 4

Number of elements of order 5, when |x| = 5 and |y| = 1 are: φ(5).φ(1) = (4).(1) = 4

Number of elements of order 5, when |x| = 5 and |y| = 5 are: φ(5).φ(5) = (4).(4) = 16

Therefore, Total = 4+4+16 = 24 elements of order 5

Answer 2

We determine the number of elements of order 5 in Z25 ⊕ Z5.

By Theorem: we may count the number of elements (a, b) in Z25 ⊕ Z5 with the property that

5 = | (a, b)| = LCM (|a|, |b|). Clearly this requires that either |a| = 5 and |b| = 1 or 5, or |b| = 5 and |a| = 1 or 5.

We consider two mutually exclusive cases.

Case 1: |a|=5 and |b| = 1 or 5.

Here there are four choices for a (namely 5, 10, 15 and 20) and five choices for b.

This gives 20 elements of order 5.