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___PPT9/0KNM4L?%LKNormal Subgroups $Definition: Let (G, *) be a group and let K be a subgroup of G. Then K is said to be a normal subgroup if gK = Kg for every g G.
Remark: Note that the definition requires gK = Kg as sets; this is not the same as saying that g * k = k * g for every k K and for every g G.
Alternative versions of the definition:
a) gKg 1 = {gkg 1: k K} = K for every g G
or
b) g 1Kg = K for every g G.
In practice, it suffices to show gKg 1 K for every g G.=
Notation: K D GVA" Z5" Z" Z!" ZO" ZX*= ' bkBj+ONExamples of Normal Subgroups ${e} and G are normal subgroups for every group G.
In an abelian group G, every subgroup H is normal since here g * h = h * g for every h H and for every g G.
If K G and [G:K] = 2, then K is a normal subgroup. In particular, the subgroup <(123)> is a normal subgroup of S3.
However, not all subgroups are normal, for example, <(12)> is not a normal subgroup of S3. Ju" [86NMHomomorphisms $
~Definition: Let (G, *) and (H, ) be groups,
and let r: G H be a function such that r (a * b ) = r (a) r (b) for all a , b G. Then r is said to be an homomorphism from G to K.
Remark: this is a generalization of isomorphism, i.e. a composition preserving map (not necessarily bijective).
Definition: Let (G, *) and (H, ) be groups, and let r: G H be a homomorphism. Then:
the image or range of r is defined as Im (r) = {h H: h = r (g) for some g G}
the kernel of r is defined as ker (r) = {g G : r (g) = e H, where e is the identity element of H}0" " " " ?]> HGML$Essential Properties of Homorphisms %%$ Proposition 14: Let (G, *) and (H, ) be groups, and let r: G H be a homomorphism. Then:
r (e) = e and r (x 1) = r (x) 1 for all x G.
Im (r) is a subgroup of H.
Ker (r) is a normal subgroup of G.
r is injective if and only if Ker (r) = {e}
Furthermore, if Im (r) is a finite group, then [G: Ker (r)] = | Im (r) |.
Proof of i., ii., iii. is left as an exercise.
Example: The mapping q: (Z, +) (Zn , ) given by q (a) = a (mod n) = [a]n for all integers a Z is a homomorphism. Im (q) = Zn, and Ker (q) = nZ .^" " " PPv/& $$?!
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___PPT9/0KNM4L?%LKNormal Subgroups $Definition: Let (G, *) be a group and let K be a subgroup of G. Then K is said to be a normal subgroup if gK = Kg for every g G.
Remark: Note that the definition requires gK = Kg as sets; this is not the same as saying that g * k = k * g for every k K and for every g G.
Alternative versions of the definition:
a) gKg 1 = {gkg 1: k K} = K for every g G
or
b) g 1Kg = K for every g G.
In practice, it suffices to show gKg 1 K for every g G.=
Notation: K D GVA" Z5" Z" Z!" ZO" ZX*= ' bkBj+ONExamples of Normal Subgroups ${e} and G are normal subgroups for every group G.
In an abelian group G, every subgroup H is normal since here g * h = h * g for every h H and for every g G.
If K G and [G:K] = 2, then K is a normal subgroup. In particular, the subgroup <(123)> is a normal subgroup of S3.
However, not all subgroups are normal, for example, <(12)> is not a normal subgroup of S3. Ju" [86NMHomomorphisms $
~Definition: Let (G, *) and (H, ) be groups,
and let r: G H be a function such that r (a * b ) = r (a) r (b) for all a , b G. Then r is said to be an homomorphism from G to K.
Remark: this is a generalization of isomorphism, i.e. a composition preserving map (not necessarily bijective).
Definition: Let (G, *) and (H, ) be groups, and let r: G H be a homomorphism. Then:
the image or range of r is defined as Im (r) = {h H: h = r (g) for some g G}
the kernel of r is defined as ker (r) = {g G : r (g) = e H, where e is the identity element of H}0" " " " ?]> HGML$Essential Properties of Homorphisms %%$$Proposition 14: Let (G, *) and (H, ) be groups, and let r: G H be a homomorphism. Then:
r (e) = e and r (x 1) = r (x) 1 for all x G.
Im (r) is a subgroup of H.
Ker (r) is a normal subgroup of G.
r is injective if and only if Ker (r) = {e}
Furthermore, if Im (r) is a finite group, then [G: Ker (r)] = | Im (r) |.
Proof of i., ii., iii. is left as an exercise.
Example: The mapping q: (Z, +) (Zn , ) given by
q (a) = a (mod n) = [a]n for all integers a Z is a homomorphism. Im (q) = Zn, and Ker (q) = nZ .2^" " d" f" PPv/( $$?!
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