presumably you mean the integers mod 55 under addition.

in this group, a "power" of 10, 10^k, is 10 added k times:

10^2 = 10 + 10

10^3 = 10 + 10 + 10

(etc.)

so we will write 10^k as k*10.

the most direct route is to show that k = 11, is the smallest positive k that will work.

1*10 = 10

2*10 = 20

3*10 = 30

4*10 = 40

5*10 = 50

6*10 = 5 (remember, we are working mod 55)

7*10 = 15

8*10 = 25

9*10 = 35

10*10 = 45

11*10 = 0 (110 = 0 mod 55, since 110 = (55)(2) + 0).