Let Z_n be the group of integers {0, 1, 2, ..., n-1} with addition modulo n as the group operation. The number of elemen...

Question

Let Z_n be the group of integers {0, 1, 2, ..., n-1} with addition modulo n as the group operation. The number of elements in the group Z_2 x Z_3 x Z_4 that are their own inverses is ___________.

Accepted Answer

There are 4 self-inverse elements in Z_2 x Z_3 x Z_4. Each group Z_n uses addition modulo n, so the identity is 0 and the inverse of x is the value that adds with x to give 0. In the product group an element is a triple (x, y, z), and it is its own inverse exactly when each coordinate is its own inverse in its own Z_n. For addition mod n, x is self-inverse means 2x is 0 mod n. Counting how many such x exist in each factor and multiplying gives the answer.