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MATH 113 Lecture Notes - Lecture 16: Normal Subgroup, Lincoln Near-Earth Asteroid Research, Coset

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16 Oct 2014

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Def: (missed) if k is a subgroup of g, then g is normal if for all g g we have gk = kg. In other words, if the left coset = right coset. Def: suppose a homomorphism $: h -> g. then ker $ = {h h | = e a} So, a value of h that leads to the identity (i. e. 0 vector) damn linear algebra. Def: suppose h is a normal subgroup of g (looks like a triangle on a line). G/h (the factor group of g by h) The operation is g1h * g2h = g1g2h. Basically, we need to show that * doesn"t depend on the choice of g1 and g2. Want to prove: given gh = g"h and g1h = g1h", then gh*g1h = g"h * g1"h. We have to show that gg1h is a subset of g"g1"h, and vice versa.

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