Show that the multiplication table for the 4-group {I, a, b, c\} given in Problem 2.9 follows from the algebraic requirements \(a^{2}=b^{2}=I\) and \(a b=b a=c\).

Data from Problem  2.9

Show that the group {e, a, b, c\} with multiplication table (b) below, is in one to one correspondence with the geometrical symmetry operations on figure (a) below

This is called the 4-group or dihedral group \(\mathrm{D}_{2}\). Show that \(\mathrm{D}_{2}\) has three subgroups, \(\{e, a\},\{e, b\}\), and \(\{e, c\}\), each isomorphic to the cyclic group \(\mathrm{C}_{2}\).

Transcribed Image Text:

(a) D2 e Ce DDC a b c b COD b e e a (b) a a e b b C e a DO C b a