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Groups and Group Actions: Lecture 8

In which we prove Lagrange’s theorem, and deduce many interesting results as a consequence. Lemma 33 (Coset equality test): Let  be a subgroup of a group .  Take , .  Have  if and only if .  For one...

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Groups and Group Actions: Lecture 8.5

In which we wonder what a non-integer lecture is anyway. OK, of course there isn’t a lecture 8.5.  But I promised that I’d post with questions for you to consider before lecture 9, so here is that...

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Groups and Group Actions: Lecture 9

In which we think about homomorphisms, and wonder how many genuinely different groups there are with small orders.  Definitions of a homomorphism, isomorphism and automorphism. Proposition 41: Let ,...

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Groups and Group Actions: Lecture 10

In which we think some more about homomorphisms, and meet normal subgroups. Proposition 45: Let  be a homomorphism.  Then there is  with  for all .  We proved this by defining  and then using the fact...

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Groups and Group Actions: Lecture 11

In which we meet and explore quotient groups. Definition of the centre of a group. Proposition 51: Let  be a group.  Then .  The proof is an exercise. Proposition 52: Let  be a group, let  be a...

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Groups and Group Actions: Lecture 12

In which we meet group actions. Definition of a left action of a group on a set. Definition of a right action of a group on a set. Definition of the orbit and stabiliser of an element of a set under an...

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Groups and Group Actions: Lecture 13

In which we explore the Orbit-Stabiliser Theorem. Proposition 58: Let  be a group acting on a set .  Take , with  and  lying in the same orbit.  Then  and  are conjugate: there is  with .  We noted...

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Groups and Group Actions: Lecture 14

In which we explore groups of order  and encounter Cauchy’s Theorem. Lemma 62: Let  be a prime, let  be a group of order .  Then  is Abelian.  We saw last time that the centre of , , is non-trivial, so...

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Groups and Group Actions: Lecture 15

In which we meet the Orbit-Counting Formula Definition of  for an element  of a group  acting on a set. Theorem 65 (Orbit-Counting Formula): Let  be a finite group acting on a finite set .  Then .  We...

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Groups and Group Actions: Lecture 16

In which we meet Cayley’s Theorem and reach the end of this particular adventure, but catch a glimpse of far-off lands still to be explored. Theorem 68: Let  be a group, let  be a set. Given a left...

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