Group_Definitions - Definitions for Geometric Groups...

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Definitions for Geometric Groups Definition: Let G be a nonempty set with a binary operation *. G is a group under * in case: (i) If a, b are elements of G, then a*b is an element of G. (Closure) (ii) If a, b, c are any three elements of G, then (a*b)*c = a*(b*c). (Associative) (iii) There exists an identity element e in G such that g*e = e*g = g for any element g in G. (Identity) (iv) If g is an element of G, then there exists an inverse element for g, denoted g –1 , such that g * g –1 = g –1 * g = e. (Inverse property). (Examples: {all integers} under addition, {all non-zero fractions} under multiplication, {all symmetries of a polygon in the plane} under “followed by” Definition: Let G be a group with binary operation * and H a nonempty subset of G. Then H is a subgroup of G if H is also a group under the operation *. Example: { all multiples of 2} is a subgroup of {all integers} if the operation is addition. {all multiples of 2} isn’t a subgroup of {all integers} under addition, if the operation we are considering for {all multiples of 2} is multiplication instead of addition.
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