# Existence of element of order 51 (TIFR 2013 problem 9)

Question:

True/False?

There is an element of order $$51$$ in $$\mathbb{Z_{103}^*}$$

Hint: 103 is prime.

Discussion: Since for $$p$$-prime, $$\mathbb{Z_p}$$ is a field with addition modulo p and multiplication modulo p as the first and second binary operations respectively, and for a finite field $$\mathbb{F}$$, $$\mathbb{F}^*$$ forms a cyclic group with respect to multiplication. Hence $$\mathbb{Z_p}^*$$ forms a cyclic group.

We have $$\mathbb{Z_{103}^*}$$, a cyclic group or order $$103-1=102$$.

For a cyclic group of order $$n$$, if $$d|n$$ then there exists an element of order d.

Since $$51|102$$, there exists an element of order $$51$$ in $$\mathbb{Z_{103}^*}$$.

Alternative: By Sylow’s first theorem, there exists elements of order 3&17 in $$\mathbb{Z_{103}^*}$$. ( $$|\mathbb{Z_{103}^*}|=102=2*3*17$$)

Since the group $$\mathbb{Z_{103}^*}$$ is abelian, the group and 3,17 are co-prime we get an element of order $$3*17=51$$ (by simply multiplying the element of order $$3$$ to the element of order $$17$$).