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1. Defined kernels using "hom" and "group_bvd": f is a homomorphism of g1 into g2. The elements of g1, which are mapped into the identity of g2, form a subgroup h of g called the kernel of f. 2. Added and proved the following properties of homomorphisms: i. Let f: G -> G' be a group homomorphism of G into G'. If H is a subgroup of G, then f[H] is a subgroup of G'. ii. Let f: G -> G' be a group homomorphism of G into G'. If H is a subgroup of G', then the inverse image of H is a subgroup of G. 3. Added and proved the following properties of kernels: i. Let f: G -> G' be a group homomorphism of G into G'. The kernel of f is a subgroup of G. ii. Let f: G1 -> G2 be a group homomorphism of G1 into G2. Let H be the kernel of f. For any a in G1, the set { x in G1 | f(x) = f(a) } is the left coset aH of H. iii.Let f: G1 -> G2 be a group homomorphism of G1 into G2. Let H be the kernel of f. For any a in G1, the set { x in G1 | f(x) = f(a) } is also the right coset Ha of H. iv. A group homomorphism f: G1 -> G2 is a one-to-one map if and only if Ker(f) = {id(g1)}. v. The kernel of a homomorphism f: G1 -> G2 is a normal subgroup of G1. Tactics: kernelSubgroupT, kernelLcosetT, kernelRcosetT, kernelNormalSubgT Note: The property that "A group homomorphism f: G1 -> G2 is a one-to- one map if and only if Ker(f) = {id(g1)}." can also be written into a tactics when necessary.
Changed def of homomorphisms and reproved all rules; Added tactics "homIdT" and "homInvT". Not finished yet. Will do the rest after fixing errors in subgroups and cylic subgroups.
Defined homomorphism for groups. A map f of a group G into a group G' is a homomorphism if f(a * b) = f(a) * f(b). Proved some properties of homomorphism with the definition. Since I would redo groups soon, this homomorphism file is not complete.
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