Page 74 - DMTH403_ABSTRACT_ALGEBRA

Unit 5: Normal Subgroups

For example, if G is a commutative group, then Notes

x y xy = x xy y = e  x, y  G.  [G, G] = {e}.
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Theorem 6: Let G be a group. Then [G, G] is a normal subgroup of G. Further, G/[G, G] is
commutative.
Proof: We must show that, for any commutator x y xy and for any g  G, g (x y- xy)g  [G,G].
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Now g (x y- xy)g = (g xg) (g yg) (g xg) (g yg)  [G, G].
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 [G, G]  G.
For the rest of the proof let us denote [G, G] by H, for convenience.
Now, for x, y  G,
HxHY = H y Hx  Hxy = Hyx  (xy) (yx)  H
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Thus, since xy x y  H  x, y  G, HxHy = HyHx  x, y  G. That is, G/H is abelian.
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Note We have defined the quotient group G/H only if H  G. But if H  G we can
still define G/H to be the set of all left (or right) cosets of H in G. But, in this case G/H will
not be a group.
Remark: If H is a subgroup of G, then the product of cosets of H is defined only when H  G.
This is because, if HxHy = Hxy  x, y  G, then, in particular,

Hx Hx = Hx x =He = H  x  G.
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Therefore, any h  H, x hx = ex hx  Hx Hx= H.
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That is, X Hx  H for any x E G.
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 H  G.
Self Assessment

1. Every subgroup of a ................... group is a normal subgroup.
(a) associative (b) large

(c) cyclic (d) commutative
2. g Hg = ................... if h  H where H is a subgroup of G.
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(a) g hg (b) gh g
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(c) gh g -1 (d) ghg
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3. The group of G is of order ................... is called dihedral group.
(a) 6 (b) 7
(c) 8 (d) 9
4. Every group of index ................... is normal.
(a) 4 (b) 5
(c) 3 (d) 2

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