Answer to Question #133614 in Abstract Algebra for PRATHIBHA ROSE C S

Question #133614
Find the order of all elements of S3
1
Expert's answer
2020-09-24T15:25:54-0400

The order of an element a of a group is the smallest positive integer "m" such that "a^m = e" (where "e" denotes the identity element of the group, and "a^m" denotes the product of "m" copies of "a" ). If no such "m" exists, "a" is said to have infinite order.


All elements of finite groups have finite order.


The order of a group "G" is denoted by "ord(G)" or "|G|" and the order of an element a by "ord(a)" or "|a|" .

The symmetric group "S_3" has the following multiplication table.



This group has six elements, so "ord(S_3) = 6" .


By definition, the order of the identity "e" is "1" . Each of "s, t," and "w" squares to "e" , so these group elements have order 2. Completing the enumeration, both "u" and "v" have order 3, for "u^2 = v" and "u^3 = vu = e" , and "v^2 = u" and "v^3 = uv = e".


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