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Unformatted text preview: Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Chapter 2: Subgroups Keith E. Emmert Tarleton State University February 16, 2010 Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Overview Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Definition of a Subgroup Definition 1 Let G be a group. The subset H of G is a subgroup of G if H is nonempty and H is closed under products and inverses. In this case we write H G . Example 2 1. Clearly, the operation is a subgroup is a transitive operation. 2. Z Q R with the operation addition. 3. Let G be a group. We have the trivial subgroup { 1 } G and we also note that G G . 4. The set of even integers is a subgroup of the set of integers (under addition). Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group A Simple Test Proposition 3 A subset H of a group G is a subgroup if and only if 1. H negationslash = 2. for all x , y H , xy 1 H. Furthermore, if H is finite, then it suffices to check that H is nonempty and closed under multiplication. Proof: Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Lagranges Theorem Theorem 4 If G is a finite group and H is a subgroup of G, then  H  divides  G  . Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Homework 2.1 Pages 48 49 1, 2, 4, 5, 8, 9, 10, 12 Chapter 2: Subgroups Keith E. Emmert Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Overview Definition and Examples Centralizers and Normalizers, Stabilizers and Kernels Cyclic Groups and Cyclic Subgroups Subgroups Generated by Subsets of a Group The Lattice of subgroups of a Group Chapter 2: Subgroups...
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