3330_Notes_4o6_fill

# 3330_Notes_4o6_fill - Math 3330 Section 4.6 Page 1 of 8...

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Math 3330 Section 4.6 Page 1 of 8 Math 3330 Section 4.6 Quotient Groups Start with what we have done in the last two days 1. Products of Subsets Let A and B be nonempty subsets of the group G. Define the product of AB to be the subset { } |, f o r s o m e , AB x G x ab a A b B =∈ = 2. Normal Subgroups – H is a normal subgroup of G if xH = Hx for all x in G. 3. HH = H 4. Properties of Set Products - A(BC) = (AB)C - B = C implies AB = AC and BA = CA for all A - Product not commutative - AB = AC does not imply B = C - gA = gB implies A = B These properties give us the ability to treat the set of cosets (this is a set of sets!) as an algebraic system with set products as the operation. Theorem: Group of Cosets Let H be a normal subgroup of G. Then the cosets of H in G form a group with respect to the product of subsets. Proof:

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Math 3330 Section 4.6 Page 2 of 8 Definition: Quotient Group If H is a normal subgroup of G, the group G/H that consists of the cosets of H in G is called the quotient group or factor group of G by H.

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## This note was uploaded on 08/10/2011 for the course MATH 3330 taught by Professor Flagg during the Spring '11 term at University of Houston.

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3330_Notes_4o6_fill - Math 3330 Section 4.6 Page 1 of 8...

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