Question

Suppose G has order 4 but contains no element of order 4.

Explain why every nonidentity element of G has order 2.

Short answer

It is explained why every non-identity element of G has order 2.

Step-by-step solution

Referring to the result of Part (a) of this Exercise

• No element of G has order 3

  Given that G has order 4, but contains no element of order 4.

Explanation

As the order of G is 4, no element of G can have order more than 4 and we have already proved in part (a) that no element of G has order 3.

Therefore, there is only one element with order 1, which is the identity element.

Hence, we can conclude that every non-identity element of G must have order 2.