Subgroups
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Subgroups
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Suppose [Equation goes here - download the original to see it.] is a group - that is, G is a set of elements and is a binary operation defined on that set and such that the four group axioms (Identify, Inverses, Closure, Associativity) are satisfied. Suppose, also, that H is a proper subset of G, that is [Equation goes here - download the original to see it.]. Then H will be a subgroup of G if the group of axioms also apply to H under operation . That is if also satisfies the group of axioms of identify, inverses, closure and associativity. We prove that [Equation goes here - download the original to see it.]is a subgroup of [Equation goes here - download the original to see it.]by verifying the group properties for H. However, if is associative on G then must be associative on H, because H is a subset of G. So to show a group is a subgroup we have to check the three properties of (1) Identify, (2) inverses and (3) Closure.To illustrate the subgroup property, consider the symmetry group S3 of the equilateral triangle. The symmetries are[Equation goes here - download the original to see it.]S3 has combination table[Diagram goes here - download the original to see it.]As the partition lines indicate, the set of rotations {I, R1, R2}Is a subgroup of S3To verify that this is a group and hence is a subgroup of S (a)Identity[Equation goes here - download the original to see it.] (b)Inverses[Equation goes here - download the original to see it.] (c)Closure The set is closed. When one group is a subgroup of another we can use the symbol [Equation goes here - download the original to see it.] or more simply [Equation goes here - download the original to see it.] The set of reflections {Q1 Q2 Q3}is not a subgroup of S3. The set is closed and each element is its own inverse, but the set does not contain the identity.
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Contents of
Subgroups
1 Subgroups
2 The null set
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