FOR THIS POST ONLY PROVE PART 2 (the 4th or lattice isomorphism theorem) Let G b

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FOR THIS POST ONLY PROVE PART 2



(the 4th or lattice isomorphism theorem) Let G be a group and N be a normal subgroup of G. Then there is a bijection from the set of subgroups of A of G which contain N onto the set of subgroups of A/N of G/N. In particular, every subgroup of G/N is of the form A/N for some subgroup A of G containing N (namely the preimage in G under the natural projection homomorphism from G to G/N). This bijection has the following properties: For all A and B that are subgroups of G with N a subgroup of A and N a subgroup of B.


1) A subgroup of B if and only if A/N subgroup of B/N

2) if A subgroup of B then |B:A|=|B/N:A/N|

3) /N =

4) (A intersected with B)/N = A/N intersected with B/N

5) A is a normal subgroup of G if and only if A/N is a normal subgroup of G/N

Explanation / Answer

We refer [2] for detail information on group theory.

A nonempty set R with two binary operations + and . is called a right near ring if (R, +) is not necessarily abelian group, (R,.) is a semi group and multiplication is right distributive over addition (x+y). z = xz+yz for all x,y,z in R.

The most common example of a near ring is (M(G),+, . ) where M(G) is the set of all mappings from a group G to itself, addition is defined point wise and multiplication is composition of maps.

If there is a near ring homomorphism from R to M(G), then G is called a (left)

R-module. is called a representation of R which is faithful if ker={0}. We write rg for (rg) for all r in R and g in G.A subgroup H of G is called an R-submodule of G if RHÍH. A normal subgroup (H,+) of G is called an R-ideal of G if r(g+h)-rg H for all g in G, h in H and r in R.

A mapping from an R-module G to another R-module K is called an R-module homomorphism or R-homomorphism if is a group homomorphism from (G,+) to (K,+) and (rg) = (r)g for all g in G and r in R.

Let be an R-homomorphism from an R-module G to an R-module K. Then G is an R-submodule of K and kernel of is an R-ideal of G. Let N be an R-ideal of G,

G/N = {g + N / g G} is the quotient R-module of G by N.

Let G be an R-module .If H is an R-submodule(R-ideal) of G and N is an R-ideal of G, then N+H is an R-submodule(R-ideal) of G.

Let G be an R-module, H an R-submodule of G and K an R-submodule(R-ideal) of G.Then HK is an R-submodule(R-ideal) of H.

We are now in a position to state the well-known three isomorphism theorems for near ring modules.

The first isomorphism theorem for near-ring modules:

Let G and K be R-modules. be an R-homomorphism from G to K with kernel N.Then

G / N @R (G).

The second isomorphism theorem for near-ring modules:

Let G be an R-module, N an R-ideal and H be an R-submodule of G. Then N+H is an

R-submodule of G and N+H / N @R H / HN.

The third isomorphism theorem for near-ring modules:

Let be an R-epimorphism from the R-module G to the R-module K with kernel N.Then there is a one-one correspondence between R-submodules (R-ideals) of G containing N and R-submodules(R-ideals) of K.This correspondence preserves and reflects inclusions. Furthermore if MÊN, where M is an R-ideal of G then (G/N) / (M/N) @R G/M.

It turns out that an analogous result of the fourth isomorphism theorem for ring modules is also true for near ring modules.

The fourth isomorphism theorem for near-ring modules:

Let N be a submodule of R-module G.There is a bijection between the submodules of G which contain N and the submodules of G/N.

The correspondence is given by A A/N for all AÊN

The correspondence commutes with the process of taking sums and intersections.

Proof:

The natural projection map : AA/N defined by (a)= a + N and the fact that every submodule N of an R-module is normal, together with the group theoretical results, prove what we claim.

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