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133 строки
2.9 KiB
Plaintext
133 строки
2.9 KiB
Plaintext
Alexandrov - Groups Theory Intro.djvu
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Page 33.
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(ab)^-1 = b^-1*a^-1
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ab(b^-1a^-1) = a(bb^-1)a^-1 = aa^-1 = 1
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Page 85.
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Group element b^-1ab is called a "transformation of a using b".
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When does the following holds???
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a = b^-1ab
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Iff (if and only if) the elements commute:
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ab = ba
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Multiplying left by b^-1 having:
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b^-1ab = a
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Lema 1 (Conjugation elements symmetry)
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If elt-b is a transformation of elt-a using elt-c
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b = c^-1ac
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Then elt-a is a transformation of elt-b uscing elt-(c^-1):
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cbc^-1 = a
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a = (c^-1)^-1bc
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Def (Conjugate elements)
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If one group element is a transformation of the other - they are called conjugant.
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Lema 2 (Conjugate elements transitivity).
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If elt-a is adjoint with elt-b, which is adjoint with elt-d,
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then elt-a is ajdoint with elt-d.
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b = c^-1ac
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b = e^-1de
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So,
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c^-1ac = e^-1de
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a = (ce^-1)d(ec^-1) = { p. 33.} = (ec^-1)^-1d(ec^-1)
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In other words, a is a transformation of d usint elt-(ec^-1).
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Lema 3 (Adjoint elements reflexivity)
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Every elt-a is adjoint to itself:
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a = 1^-1a1
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Theorem 1 (Conjugacy classes)
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Conjugant relation splits group G to the set of equivalence-classes.
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Note: Class of neutral el-t 1 consists only of 1-itself:
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a^-1*1*a = 1 for all a
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Def (subgroup transformation)
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Conside H - subgroup of G.
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Let's fix some el-t b of G.
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We call group transformation b^-1Hb a set of
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b^-1Hb = { b^1xb | x of H }
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We can prove (p. 91) that b^-1Hb is indeed a group.
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It is obvious that H is isomorphic to b^-1Hb.
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Theorem 2.
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Transformation of subgroup H (of group G) using el-t b of G
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is a subgroup of G, isomorphic to H.
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Note 1: If H is a subgroup of commutative/abelian group G.
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then subgroup transformation of H using any b of G is H itself
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b^-1xb = {commutativity} b^-1bx = x
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Note 2: For every group G and subgroup H, if b belongs to H, then
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b^-1Hb = H
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since b^-1xb belongs to H
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Note 3: If subgroup H_2 is a transformation of H_1 using b,
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then H_1 is a transformation of H_2 using b^-1
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Def (Conjugative subgroups)
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Two subgroups of group G are called conjugative if one of them is a transformation of another.
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Conjugation relation splits subgroups to the equivalence-classes.
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Invariant/Normal subgroup
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Def 1. If subgroup H of G has no conjugative subgroup (except itself), then it is called
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normal or invariant subgroup.
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In other words - transformation of any element of H, using any element of G, belongs to H.
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Example: any subgroup of commutative-abelian group is normal/invariant.
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Homomorphisms and Kernel
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Page 97...
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Page 99.
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Kernel of homeomorhic group morphism:
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f : A -> B
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f(ab) = f(a) f(b)
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if the set
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a of A, such that
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f(a) = 1_B
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In other words, Kernel(f) = f^-1(1_B).
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Page 111 - Factor groups...
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Page 113.
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If U is some normal/invarian gubgroup of G, the set of equivalence-classes V
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is a group itself (U is a neutral element).
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Group V is called a factor-group of G.
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