NoteNextra-origin/content/Math4302/Math4302_L15.md

Math4302 Modern Algebra (Lecture 15)

Group

Normal subgroup

Suppose H\leq G, then the following are equivalent:

1. aH=Ha for all a\in G
2. aHa^{-1}= H for all a\in G
3. aha^{-1}\subseteq H for all a\in G

Then H\trianglelefteq G

Tip

If H\leq G and if aH is a right coset, then aH=Ha.

Reason: If aH=Hb for some b\in G, then a\in aH, so a\in Hb but a\in Ha, so Hb=Ha.

Example

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

For example, if \det:GL(n,\mathbb{R})\to \mathbb{R}-\{0\} is a homomorphism, then

H=\ker(\det)=\{A\in GL(n,\mathbb{R})|\det(A)=0\}=SL(n,\mathbb{R})\trianglelefteq GL(n,\mathbb{R})

Factor group

Consider the operation on the set of left coset of G, denoted by S. Define

(aH)(bH)=abH

Condition for operation

The operation above is well defined if and only if H\trianglelefteq G.

Proof

First, suppose H\trianglelefteq G, and $aH=a'H$m and bH=b'H, we want to show that abH=ab'H.

It is enough to show that (ab)^{-1}a'b'=b^{-1}a^{-1}a'b'\in H.

aH=a'H\implies a^{-1}a'\in H, and bH=b'H\implies b^{-1}b'\in H. Note that by proposition of normal group, gHg^{-1}\subseteq H for any g\in G, so let g=b^{-1}, h=a^{-1}a.

Therefore b^{-1}(a^{-1}a')(b^{-1})^{-1}=b^{-1}a^{-1}a'b\in H, since b^{-1} b'\in H, then b^{-1}a^{-1}a'b'\in H.

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Conversely, suppose this operation is well defined, then we show that ghg^{-1}\in H for any g\in G, h\in H.

Note that hH=eH, the well-defineness implies that (hH)(g^{-1}H)=(eH)(g^{-1}H)=g^{-1}H. So ghg^{-1}\in H. (add g on the left)

aH=bH\iff a^{-1}b\in H, or equivalently aH=bH\iff b^{-1}a\in H.

Theorem for operation over left coset

If H\trianglelefteq G, the set of left coset of G is a group under the operation defined above.

Proof

This operation is well defined by condition above.

• Identity: eH=H
• Inverse: (aH)^{-1}=a^{-1}H
• Associativity: (aH bH)cH=aH(bH cH)=abcH

Such group is called the factor group of G by H.

(Non) Example of factor group

Recall from previous lectures, G=S_3 with H=\{e,\tau_1\}, with \tau_1=(12), \tau_2=(23), \tau_3=(13).

• \{e,\tau_1\}=\tau_1 H=H
• \{\tau_2,\rho_2\}=\tau_2 H=\rho_2 H
• \{\tau_3,\rho\}=\tau_3 H=\rho H

And (\tau_2 H)(\tau_3 H)=\tau_2 \tau_3 H=\rho H.

However, if we take \rho^2\in \tau_2 H, and \rho\in \tau_3 H, \rho^2\rho =e. This is not in \rho H.

This is not well defined since H is not normal.

Definition of factor (quotient) group

If H\trianglelefteq G, then the set of cosets with operation:

(aH)(bH)=abH

is a group denoted by G/H. This group is called the quotient group (or factor group) of G by H.

Example

5\mathbb{Z}\trianglelefteq \mathbb{Z}, the cosets are 5\mathbb{Z}, 1+5\mathbb{Z}, 2+5\mathbb{Z}, 3+5\mathbb{Z}, 4+5\mathbb{Z}.

Here 5\mathbb{Z} is the identity in the factor group.

And \mathbb{Z}/5\mathbb{Z}\simeq \mathbb{Z}_5