NoteNextra-origin/content/Math4302/Math4302_L18.md

Math4302 Modern Algebra (Lecture 18)

Groups

Factor group

Suppose G is a group, and H\trianglelefteq G, then G/H is a group.

Recall from last lecture, if \phi:G\to G' is a homomorphism, then G/\ker(\phi)\simeq \phi(G)\leq G'.

Example (continue from last lecture)

\mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle\simeq \mathbb{Z}

Take \phi(a,b)=a-b, this is a surjective homomorphism from \mathbb{Z}\times\mathbb{Z}/\langle (1,1)\rangle to \mathbb{Z}

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\mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle\simeq \mathbb{Z}

where \langle (2,1)\rangle=\{(2b,b)|b\in \mathbb{Z}\}

Take \phi(a,b)=a-2b, this is a surjective homomorphism from \mathbb{Z}\times\mathbb{Z}/\langle (2,1)\rangle to \mathbb{Z}

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\mathbb{Z}\times\mathbb{Z}/\langle (2,2)\rangle

This should also be a finitely generated abelian group. (\mathbb{Z}_2\times \mathbb{Z} actually)

Take \phi(a,b)=(a\mod 2,a-b)

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More generally, for \mathbb{Z}\times \mathbb{Z}/\langle (a,b)\rangle.

This should be \mathbb{Z}\times \mathbb{Z}_{\operatorname{gcd}(a,b)}

Try to do section by gcd.

• If G is abelian, N\leq G, then G/N is abelian.
• If G is finitely generated and N\trianglelefteq G, then G/N is finitely generated.

Definition of simple group

G is simple if G has no proper (H\neq G,\{e\}), normal subgroup.

Tip

In general S_n is not simple, consider the normal subgroup A_n.

Example of some natural normal subgroups

If \phi:G\to G' is a homomorphism, then \ker(\phi)\trianglelefteq G.

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The center of G: Z(G)=\{a\in G|ag=ga\text{ for all }g\in G\}

Z(G)\trianglelefteq G.

• e\in Z(G).
• a,b\in Z(G)\implies abg=gab\implies ab\in Z(G).
• a\in Z(G)\implies ag=ga\implies a^{-1}\in Z(G).
• If g\in G, h\in Z(G), then ghg^{-1}\in Z(G) since ghg^{-1}=gg^{-1}h=h.

Z(S_3)=\{e\}, all the transpositions are not commutative, so Z(S_3)=\{e\}.

Z(GL_n(\mathbb{R}))? continue on friday.