# Theorems on the Order of an Element of a Group

__Theorem 1__**:** The order of every element of finite group is finite.

__Proof__**:** Let be a finite group and let , we consider all positive integral powers of , i.e.

Every one of these powers must be an element of . But is of finite order. Hence these elements cannot all be different. We may therefore suppose that

Now

Since is a positive integer,

hence there exists a positive integer such that.

Now, we know that every set of positive integers has at least number. It follows that the set of all those positive integer such that has a least member, say , thus there exists a least positive integer such that , showing that the order of every element of a finite group is finite.

__Theorem 2__**: **The order of an element of a group is the same as that of its inverse .

__Proof__**:** Let and be the orders of and respectively.

Then, and

Now

Also

Because

Now and

If the order of is infinite, then the order of cannot be finite. Because

is finite. Therefore if the order of infinite, then the order of must also be infinite.

__Theorem 3__**:** The order of any integral power of an element cannot exceed the order of .

__Proof__**: **Let be any integral power of . Let .

Now,

__Theorem 4__**:** If the element of a group is order , then if and only if is a divisor of .

__Theorem 5__**:** The order of the elements and is the same where are any two elements of a group.

__Theorem 6__**:** If is an element of order and is prime to , then is also of order .

__Corollary__**:** The order of is the same as that of where and are any elements of a group.