Page 75 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 5. If H and k are normal subgroup of group G, then so is H ................... k.
(a)  (b) 

(c) = (d) 
5.3 Summary

We discussed here:
The definition and examples of a normal subgroup.

Every subgroup of an abelian group is normal.

Every subgroup of index 2 is normal.

If H and K are normal subgroups of a group G, then so is H  K.

The product of two normal subgroups is a normal subgroup.

If H  N and N  G, then H need not be normal in G.

The definition and examples of it quotient group.

If G is abelian, then every quotient group of G is abelian. The converse is not true.

The quotient group corresponding to the commutator subgroup is commutative.

The set of left (or right) cosets of H in G is a group if and only if H  G.

5.4 Keywords

Normal Subgroup: A subgroup N of a group G is called a normal subgroup of 6 if Nx = xN  x
 G, and we write this as N  6.
Dihedral Group, D : It is the group of symmetries of a square, that is, its elements represent the
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different ways in which two copies of a square can be placed so that one covers the other.
Quotient Group: If C = Hx = Hx and C = Hy = Hy , then is C C = Hxy or is C C = Hx y ?
1
2
1
1
2
1 1
1
1
2
Actually, we will show you that Hxy = Hx y , that is, the product of cosets is well-defined.
1 1
5.5 Review Questions
1. Show that A  S .
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3
2. Consider the subgroup SL (R) = {A E GL (R) | det(A) = 1} of GL (R). Using the facts that det
2
2
2
1
(AB) = det (A) det (B) and det (A ) = , prove that SL (R)  GL (R).
–1
det(A) 2 2
3. Consider the group of all 2 × 2 diagonal matrices over R*, with respect to multiplication.
How many of its subgroups are normal.
4. Show that Z(G), the centre of G, is normal in G. (Remember that Z(G) = {x  G| xg = gx 
g  G}).
5. Show that <(2 3)> is not normal in S .
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