Definitions
Reflection: A reflection about a line, l, transforms each point, P, in the
plane in the direction of minimum distance to the line (perpendicular) and
through the line in the same direction by the same dis•
Rotation: A rotation about a point, P, transforms each other point, Q, in
the plane to a point Q’ so that it maintains the same distance to P and
angle Q’PQ is the same for all points Q.
Translation: A translative transforms every point P in the plane, to a new
point P’ in the plane, in the same direction and by the same distance.
Isometry: An isometry of the plane transforms every object in the plane so
that all distances and angles are preserved.
Group: A group is a set closed under a binary operation (like composition) such
that (1) the operation is associative (2) the set contains an identity (3)
every element of the set has an inverse.
Closure: A set is closed under a binary operation when any two elements of the set are combined under that binary operation, the result is another element of the set.
Identity: The identity element in a set is an element, e, of that set such ghat, for all elements in the set, e*a = a*e = a
Inverse: For any element a in the set, there exists an element a^-1 such that a*a^-1 = e
Group: A group is a set with a binary operation, * (e.g., composition), that
satisfies the following properties:
Closure: A set is closed under a binary operation if, when any two elements of the
set are combined under that binary operation, the result is another element of the
set.
Identity: The identity element in a set is an element, e, of that set such that, for all
elements a in the set, e*a=a*e=a.
Inverses: For any element a in the set, there exists an inverse element, a-1, such that
a*a-1 = a-1 *a= e.
Associativity: For all elements a, b, and c in the set, (a*b)*c=a*(b*c).
Subgroup: A subgroup, H, of a group, G, is a group (under the same
operation, and which contains the same identity) and is subset of the
group, G.