Homomorphisms
When we think about a group, it is naturally for us to think about the homomorphisms. We want to know all the homomorphisms of this group.
By the “First Isomorphism Theorem” (#reference 1), we know that G/Ker (Phi) ≈ Phi (G). So, if we know the kernels of the homomorphisms, we will be able to find out what each homomorphism is. So, naturally, the next question is how to find the kernels of an arbitrary homomorphism without knowing anything about the homomorphism?
A Special Propriety of Kernels
For a group G, and an arbitrary homomorphism Phi, Ker (Phi) is a subgroup of G (#reference 2). For any e ∈ Ker (Phi), any x ∈ G,
Phi(xex^(-1))= Phi(x)Phi(e)Phi(x^-1)=Phi(x)Phi(x^-1)=Phi(x*x(^-1))=1
So, for any x ∈G, e ∈ Ker (Phi), xex^(-1)∈Ker(Phi). Therefore, for the kernel of any arbitrary homomorphism has the propriety:
xKer(Phi)x^(-1) ⊆ Ker(Phi).
Therefore, whenever we found a subgroup of G has the propriety xHx^(-1) ⊆ H, we know it is a kernel of a homomorphism.
Kernels are Normal, Definition of Normal Group.
We define normal groups as follow:
A subgroup H of G is normal in G if and only if xHx^(-1) ⊆ H for all x in G.
Immediately, we know that kernels are normal. We can show that for a normal group H, xH=xH for all x in G. And if H is a subgroup of G, for all x in G, xH=xH, we can show that H is normal.
1. If xH=xH, then for any x ∈ G, h ∈ H, there exists an h’ such that xh=h’x. So,
xhx^(-1)=h’. Therefore, xHx^(-1) ⊆ H.
2. If , xHx^(-1) ⊆ H, let x= a, the aHa^(-1) ⊆ H. So, for any h ∈H, there exists a h’,
Such that ah=h’a^(-1). So, aH⊆ Ha. Similarly, let x=a^(-1), we can show Ha ⊆ aH.
Therefore, aH=Ha.
Factor Groups
Since G/Ker (Phi) ≈ Phi (G), by finding a normal group H and the factor group G/H, one can understand a homomorphism.
Theorems About Normal Groups (#reference 3)
1. G/Z Theorem
Let G be a group and let Z(G) be the center of G. If G/Z(G) is cyclic, then G is Abelian.
2. For any group G, G/Z(G) is isomorphic to Inn(G).
3. Cauchy’s Theorem for Abelian Groups
Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p.
4. Galois Theory
If K is the splitting field of some polynomial in F[x], then Gal(E/K) is a normal subgroup of Gal(E/F), and Gal(K/F) is isomorphic to Gal(E/F)Gal(E/K).
One Question
Let H be a normal subgroup of a group G and K be any subgroup of G. Then HK is a subgroup of G. (#reference 4). But can we say that if H and K are subgroups of G, and HK is also a subgroup of G, then H is a normal subgroup?
Links
http://math.colgate.edu/math320/dlantz/extras/notes11_13.pdf
http://en.wikibooks.org/wiki/Abstract_Algebra/Group_Theory/Normal_subgroups_andQuotientgroups
Normality is not transitive: http://groupprops.subwiki.org/wiki/Normality_is_not_transitive
Generator of normal group:http://math.stackexchange.com/questions/130297/generator-of-normal-group
Minimal normal subgroup (View subgroup as product of normal groups) http://planetmath.org/encyclopedia/GroupSocle.html
Number of Normal subgroups in a p-Group : http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group
http://groupprops.subwiki.org/wiki/Abelian_normal_subgroup_of_group_of_prime_power_order
Characteristic implies normal http://groupprops.subwiki.org/wiki/Characteristic_implies_normal
Normal Subgroups of Groups which are Products of Two Abelian Subgroups
http://www.jstor.org/stable/info/2038627
Reference
Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole
MA 453 Notes
Category:MA453Fall2012Walther
