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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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