Abstract Algebra/Group tables

The Group of Order 2 Here is the group table for the only group of order 2 + Z2+ 0 1 0 0 1 1 1 0 The Group of Order 3 Here is the group table for the only group of order 3 + Z3+ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 The Groups of Order 4 Here are the group tables for the only groups of order 4 The cyclic group of order 4 Two ways of documenting the same group structure Z4 + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 + × 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 To see more clearly that these two tables actually have the same group structure you'll need to rename the entries      0       maps to       1       1 maps to 2 2 maps to 4 3 maps to 3 1 + 2 = 3 maps to 2 × 4 = 3 Notice that regardless of the way we notate this group, there is an element that generates the whole group. The other group of order 4 For the following example, image the number 0 through 3 written in binary, then add the digits without any carrying. For example, 2 + 3 10 + 11 01 1 Since binary addition (without carry) is isomorphic to we view this group as being two copies of joined together. That's where the name comes from. + + 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 The Group of Order 5 + + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 Other small groups A list of groups of order 1 through 31 compiled by John Pedersen, Dept of Mathematics, University of South Florida A list of groups names and some examples of group graphs from Wolfram, makers of Mathematica. Abstract Algebra