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Always commutative - PowerPoint PPT Presentation

Let H be a normal subgroup of G, and let G/H={Hg|g ? G} For ? Hg 1 and Hg 2 ? G/H,

Theorem 6.21: Let H be a subgroup of G. H is a normal subgroup of G iff g -1 hg ? H for ? g ? G and h ? H. Proof: (1) H is a normal subgroup of G (2) g -1 hg ? H for ? g ? G and h ? H For ? g ? G , Hg?=gH. Let H be a normal subgroup of G, and let G/H={Hg|g ? G}

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Proofs Using Logical Equivalences

Proofs Using Logical Equivalences. Rosen 1.2. List of Logical Equivalences. p ?T ? p; p?F ? p Identity Laws p?T ? T; p?F ? F Domination Laws p?p ? p; p?p ? p Idempotent Laws ?(?p) ? p Double Negation Law p?q ? q?p; p?q ? q?p Commutative Laws

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Always commutative - PowerPoint PPT Presentation

Let H be a normal subgroup of G, and let G/H={Hg|g ? G} For ? Hg 1 and Hg 2 ? G/H,

Proofs Using Logical Equivalences