Lecture18 - Lecture 18 6 Abstract algebra and coding theory...

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Unformatted text preview: Lecture 18 6 Abstract algebra and coding theory (continued) 1 Groups 1 / 15 Groups XIV The lemma implies: Theorem If two cosets have one element in common then they are the same. Hence two cosets are either disjoint or the same. The group G is the disjoint union of the distinct cosets. We make the following definition: Definition The set of left cosets of a subgroup H in a group G is denoted by G / H , called G modulo H . Note that G / H is a set whose elements are sets (subsets of G ). We think of each of these elements as a point in G / H . Hence G / H is the set G / H = { a | a G } . If x is an element of a , we say that x represents the coset a . The coset of e is simply H . Two elements of G which differ by right multiplication with an element of H become the same element in G / H . 2 / 15 Groups XV The number of elements of a group G is called the order of G and is denoted by | G | (which could be infinite). The number of elements of G / H is called the index of H in G , denoted by ( G : H ). Since the group G is the disjoint union of the different left cosets and each coset has | H | elements we get: Theorem (Lagrange) For a group G with a subgroup H we have | G | = | H | ( G : H ) . Hence if G is finite, then | H | is a divisor of | G | . Corollary A group G of prime order has only the trivial subgroups G and { e } . 3 / 15 Groups XVI Certain subgroups are particularly important: Definition A subgroup H of a group G is called normal subgroup if gHg 1 H , for all g G , where gHg 1 = { ghg 1 | h H } . Note that every subgroup of an abelian group is normal, because in this case gHg 1 = gg 1 H = H . Normal subgroups are important because if we have a normal subgroup G , then the set of cosets G / H forms itself a group . This is a fundamental fact which can be used to construct many new groups....
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Lecture18 - Lecture 18 6 Abstract algebra and coding theory...

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