Quantcast
Channel: Theorem of the week
Browsing latest articles
Browse All 10 View Live

Image may be NSFW.
Clik here to view.

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...

View Article


Image may be NSFW.
Clik here to view.

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...

View Article

Image may be NSFW.
Clik here to view.

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 ,...

View Article

Image may be NSFW.
Clik here to view.

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...

View Article

Image may be NSFW.
Clik here to view.

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...

View Article


Image may be NSFW.
Clik here to view.

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...

View Article

Image may be NSFW.
Clik here to view.

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...

View Article

Image may be NSFW.
Clik here to view.

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...

View Article


Image may be NSFW.
Clik here to view.

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...

View Article


Image may be NSFW.
Clik here to view.

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...

View Article
Browsing latest articles
Browse All 10 View Live