Groups and Group Actions: Lecture 11

In which we meet and explore quotient groups.

  • Definition of the centre of a group.
  • Proposition 51: Let G be a group.  Then Z(G) \trianglelefteq G.  The proof is an exercise.
  • Proposition 52: Let G be a group, let H be a subgroup of G.
    1. Define a binary operation \ast on G/H via (g_1 H) \ast (g_2 H) = (g_1 g_2) H.  This is well defined if and only if H \trianglelefteq G.
    2. If H \trianglelefteq G, then (G/H, \ast) is a group.

    Each part was a careful check, relying on the definition of a normal subgroup.

  • Definition of a quotient group.
  • Proposition 53: Let G be a group, let H be a subset of G.  Then H is a normal subgroup of G if and only if it is the kernel of a homomorphism with domain G.  One direction was Proposition 49.  For the other, where H \trianglelefteq G so G/H is a group, we defined the quotient map \pi : G \to G/H by \pi(g) = gH, and showed that this is a homomorphism with kernel H.
  • Theorem 54 (First isomorphism theorem): Let G, H be groups, let \theta : G \to H be a homomorphism.  Then G /\ker \theta \cong \mathrm{Im}\theta, and the map \tilde{\theta} : G /\ker\theta \to \mathrm{Im} \theta given by \tilde{\theta}(g \ker \theta) = \theta(g) is an isomorphism.  We checked that \tilde{\theta} is well defined (using Proposition 47), that it is a homomorphism, that it is injective, and that it is surjective.
  • Corollary 55: Let G be a finite group, let H be a group, let \theta : G \to H be a homomorphism.  Then |G| = |\ker \theta| \cdot |\mathrm{Im}\theta|.  We used our proof of Lagrange’s theorem, which showed that if K \leq G then |G| = |G/K| \cdot |K|.

Understanding today’s lecture

Can you prove Proposition 51 (that the centre of a group is a normal subgroup of that group)?

Can you fill in all the details in the proof of Proposition 52(ii) (that (G/H, \ast) is a group)?

You could explore Corollary 55 by picking some explicit examples and seeing how it works for them.

Further reading

Tim Gowers has written a really interesting blog post about normal subgroups and quotient groups.  He’s also written about what it means to say that a function is well defined, which was very relevant for today’s lecture.

A long time ago, I wrote something about the first isomorphism theorem.  There’s also a more recent hedgehogmaths video.

If, like me, you have recently found yourself trying to remember/discover how to write \trianglelefteq or \cong in LaTeX, then you will appreciate Detexify.

Preparation for Lecture 12

How many homomorphisms are there from S_3 to C_4 \times C_2?  (We’ll do this in the lecture, but do have a go before then.)

We’re also going to meet the notion of a group action.  You could read this post by Tim Gowers to start to get a feel for what that’s about.


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