Answer to Question #287028 in Abstract Algebra for Himangani

Question

{F} 9.

A) Let U(10)={1,3,7,9} be a group under multiplication modulo 10, what is the order of group?

B) What is the order of group Z of integers under addition?

Answer 1

A) The order of integers is simply the cardinality which is the number of elements in the set.

For this case the number of elements is four.

Taking ( 4 ) modulo 10 = 4 which is the required solution.

B) Here the integers z under addition form a cyclic group.

z is generated by either 1 0r -1.

Usually the order of an element g in some group is the least positive integer n such that gⁿ = 1, if any such n exists. If there is no such n, the the order of g is defined to be infinity.

Now, for the additive group Z of integers, every non zero element has infinite order.

Here, using additive notation, we calculate the order of g which is an element of Z by looking for the least positive integer n if any such that ng = 0, if any.

But unless g = 0, there is no such n, so the order of g is infinity which is the order of group Z of integers under addition.