Groups and Group Actions Lecture 10

In which we think some more about homomorphisms, and meet normal subgroups.

  • Proposition 45: Let \theta : \mathbb{Z} \to \mathbb{Z} be a homomorphism.  Then there is n \in \mathbb{Z} with \theta(m) = nm for all m \in \mathbb{Z}.  We proved this by defining n = \theta(1) and then using the fact that 1 generates \mathbb{Z}.
  • Definition of the kernel and image of a homomorphism.
  • Proposition 46: Let G, H be groups, let \theta : G \to H be a homomorphism.  Then
    1. \ker \theta is a subgroup of G; and
    2. \mathrm{Im} \theta is a subgroup of H.

    We proved this using the subgroup test.

  • Proposition 47: Let G, H  be groups, let \theta: G \to H be a homomorphism.  Then \theta is constant on each coset of \ker \theta, and takes different values on different cosets.  We saw that \theta(g_1) = \theta(g_2) if and only if g_1 g_2^{-1} \in \ker\theta, and then used Lemma 33 to see that this is equivalent to g_1 \ker \theta = g_2 \ker \theta.
  • Corollary 48: Let G, H be groups, let \theta: G \to H be a homomorphism.  Then \theta is injective if and only if \ker \theta = \{ e_G \}.  This was immediate from Proposition 47, since e_G \in \ker \theta.
  • Definition of a normal subgroup of a group.
  • Proposition 49: Let G, H be groups, let \theta : G \to H be a homomorphism.  Then \ker\theta is a normal subgroup of G.  We already know from Proposition 46 that \ker\theta is a subgroup of G, so we just checked that if k \in \ker \theta and g \in G then g^{-1} k g \in \ker\theta.
  • Proposition 50: Let H be a subgroup of a group G with index |G/H| = 2.  Then H \trianglelefteq G.  We argued that the only (left and right) cosets of H in G are H and G \setminus H, and used this to see that gH = Hg for all g \in G.

Understanding today’s lecture

Pick some homomorphisms.  Can you identify their kernels and images?  Which homomorphisms are injective?  Which are surjective?

Pick a homomorphism \theta between two groups (pick explicit groups and an explicit homomorphism).  What are the cosets of \ker \theta?  This might help you to get a feel for Proposition 47.

Pick some subgroups of groups.  Which are normal?

Further reading

Of course Wikipedia has a page about normal subgroups, and another about the notion of a simple group.  You will learn more about simple groups if you choose to study Algebra 3 (Group Theory) next year.

MacTutor suggests that Galois was the first to recognise the significance of normal subgroups.

Preparation for Lecture 11

Can you show that if H is a normal subgroup of G then G/H (the set of left cosets of H in G) forms a group under the natural operation?

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One Response to “Groups and Group Actions Lecture 10”

  1. Groups and Group Actions Lecture 11 | Theorem of the week Says:

    […] Expositions of interesting mathematical results « Groups and Group Actions Lecture 10 […]

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