By Joan M. Holland (auth.)

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5 Use of the axioms to solve an equation relating elements in a group Let ax=b where a, b, and x are all members of a group whose identity element is e. It is often convenient to use an alternative form of this relation which isolates x. The alternative form is x=a- 1b and it is derived from the first relation by use of the group axioms as follows: ax=b, a- 1(ax)=a- 1b, every element a has an inverse a- 1 , (a- 1a) x=a- 1b, associative property, e x=a- 1b, by definition of inverse, x=a- 1b, property of the identity element.

Is this a commutative group? 9 E\1)8 A C D E6s;6E E6;:6A Reference pentagon DC DC CD C B Fig. 8 The figures above show p as a clockwise rotation of 72° about 0 applied to the reference pentagon and q as a half-turn about OA. pq=p succeeding q. The identity transformation is e. (a) Express a half-turn about OB in terms of p and q. (b) Show that qp=p4q. 9 Transforms and automorphisms In any group it is often found useful to consider the element a- 1ba where a and b are any elements and a- 1 is the inverse of a such that aa- 1=a- 1a=e, the identity element.

3 // (2) (3) (1) // (6) (4) /(5) /(6) (1) (5) (4) (3) (6((4)/(5) (2) (3) (1) (6) (2) (5) (6) (5) (6) ~~ (4) (5) ~~ (5) (4) (6) (2) (3) (1) (2) (3) (1) ~~ ~~ Definition of a Group The axioms which define a group will now be stated using the examples already discussed for illustration. A set of elements a, b, c ... together with a rule for combining any forms a group G if the following four two of them, symbolised by conditions are satisfied: * Axiom I Axiom II * b is a member of G. Associative Property: for all a, b, c in G (a * b) *c Closure: for all a, b in G, a = a *(b *c).