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Applications and Generalizations in Goursat’s LemmaApplications and Generalizations in Goursat’s LemmaCaridad Arroyo Quijano¹; Sean Eggeleston²; Boanne MacGregor³

¹University of Puerto Rico, Cayey Campus; ²Cornell College; ³Grinnell CollegeThe University of Iowa, Mathematics Department

INTRODUCTION

DISCUSSIONCONCLUSIONS

DISCUSSIONRESULTS

REFERENCES

Figure 3: Disjoint Triangle Figure 4: Isomorphism

ABSTRACT

ACKNOWLEDMENTS

Dr. Dan Anderson, Dr. Paul –Hermann Zieschang and Mr. Carlos de la Mora for their work and dedication.

Goursat’s Lemma describes subgroups of a direct product in terms of normal subgroups of the individual groups.This statement also gives an isomorphism of the form f : H12/H11→ H22/H21

where G1 and G2 are groups, Hi2 ≤ Gi, and Hi1 Hi2. We would like to generalize Goursat’s Lemma to find the subgroups of G1 × G2 × G3 × . . . × Gn. Of course, we could do this by applying the lemma recursively to G1 × ( G2 × G3 × . . . × Gn); however, this quickly grows convoluted and turns out an inelegant solution. We instead wish to generate the subgroups of G1 × . . . × Gn in terms of the subgroups of G1,G2, . . . ,Gn independently. Specifically, we examine the case where n = 3.

Before we solved this problem, we needed to considerate the Lemma restrictions: a) we need to create all the subgroups in the cross product and b) invertible functions according to the Lemma.

In order to simplify the creation of everything in the product we created these sets for

We used Goursat’s Lemma in a recursively way, , creating homomorphic projections from to and respectively as in Figure 1. According to the Lemma, there is an isomorphism and does not exist an isomorphism between

obtaining the Figure 2.

We mostly worked with finite cyclic groups, so in terms of the generators, we demonstrated that if and if then and , but we founded that for then and complicating the problem.

We tried to find a coset representative to create an isomorphism with every element of the group with its kernel. Making it harder to generalized.

If you follow the Lemma then for you could find for any pair of groups there exists isomorphic quotient groups creating a series of isomorphism defined by for

creating a series of isomorphism within , given by where , obtaning the Figure 3.

This is an interesting problem that needs more time and dedication to be solved. A next step would be finding the conjecture between the cosets representative so we could create an isomorphism that could work for all cases. Then we should check if there’s a difference for odd and even number group cases.

In the linear algebra study we would find the linear transformation of the cross product. But, in order to simplify calculations; how is this binary operation algebra? Now, this becomes an abstract algebra problem.

In order to understand this task Edouard Jean Baptiste Goursat, a Frech Mathematician, found an algorithm with two groups1, G1 and G2 , known as .

This Lemma says:

Anderson, Dan D., and Vic Camillo. “Subgroups of Direct Products of Groups, Ideals and Subrings of Direct Products of

Rings, and Goursat’s Lemma.” Contemporary Mathematics

480 (2009): 1-11. Print.

Figure 1: Recursion Figure 2: Pyramid Lemma

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•We found a series of isomorphism and homomorphism between our quotient groups applying Goursat’s Lemma like in Figure 3 and 4. •We can’t use the Lemma generalization, using isomorphism between all the quotient groups of the groups. You won’t get all the subgroups of the direct product.

• We defined as necessary two homomorphism and a isomorphism.

•We couldn't find a conjecture to make the isomorphism using the cosets representatives to create the isomorphism with everything on the set with its kernel.

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