Mathematics- Normal Subgroups, Homomorphisms, Permutation Groups Detailed Easy Notes, Study notes of Mathematics

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Mathematics- Normal Subgroups, Homomorphisms, Permutation Groups Detailed Easy Notes with Solved Exercises for BS Mathematics, MSc Mathematics, and Competitive Exams Preparation Normal Subgroups, Homomorphisms, Permutation Groups Introduction We take up a very special class of subgroups called the normal subgroups here that lead us to another class of groups called factor or quotient groups. We later take up the notion of isomorphism (a type of equality) in algebraic systems. In the end, we discuss the permutation groups. Definition: A subgroup H of a group G is called a normal subgroup of G if Ha = aH for all aeG A normal subgroup is also called invariant or self conjugate subgroup. Clearly G and {e} are normal subgroups of G and are referred to as the trivial normal subgroups. A group G # {e} is called a simple group if the only normal subgroups of G are {e} and G. Any group of prime order is simple. See theorem 25 on page 86. This group has no subgroups (let alone the normal ones) except {e} and G. It is easy to see that if H is a normal subgroup of G and K is a subgroup of G s.t.. Hc K g'Hg = g" (gH) = (¢'g) H =H. Conversely, let g'Hg=H forallgeG