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4 FINITE GROUPS AND CYCLIC GROUPS
4.1 Quarternion group Let {^ }^. then becomes a group if we define multiplication in by
. This group is called the quarternion group. This group is non-abelion and has order 8. The multiplication table for the quarternion group is as follows
4.2 Group of integers modulon Let be a fixed integer. If and are any integers, is said to be congruent to modulon (written mod if is divisible by. Examples
but Definition
Let { } and define addition in to be when the sum is divided by. Then with this operation becomes a group called the group of integers modulo under addition. This group is denoted by Example { } (^) is the group of integers modulo 4. The multiplication table for is as follows.
4.3 Cyclic groups. Let be a group and let , then the smallest positive integer such that is called the order of. If no such a number exists, is said to be of infinite order. We always take Under addition operation is denoted by times Example
- Let {^ }, the quarternion group. Find the order of Solution Since , 1 is of order 1 is order 2 , I is of order 4 , k is of order 4
Example
- Let { }, the quarternion group. Find the cyclic sub group generated by Solution 〈 〉 { } (^) i.e raise the powers until you get the identity
- Let {^ }^ be the group of integers modulo 4 under addition. Find the cyclic subgroups of generated by i) 3 ii) 2 iii) is cyclic Solutions i) 〈 〉^ {^ } ii) 〈 〉^ {^ } iii) is cyclic because it is generated by
- Consider the group of integers under addition. Find 〈 〉 Solution 〈 〉 { } Its infinite.
Theorem
Every cyclic group is abelian.
Proof Let be a cyclic group generated by. if , then there exists integers such that and .
, hence is abelian.
Theorem A subgroup of a cyclic group is cyclic.
