"Definition 1. A group G is an extension of a group A by a group B if and only if A is a normal subgroup of G and the factor group G/A is isomorphic to B. Definition 2. Two extensions G and H of A by B are called equivalent if and only if there exists an isomorphism between G and H that on A coincides with the identity automorphism and that maps onto each other the cosets of A corresponding to one and the same element of B. Consider the following example: let G be the cyclic group of order 4, that is G = {1, a, a², a³} and let H[subscript G] = {1, a²} be a normal subgroup of G. Now let V be the Klein four-group, that is, V = {1, a, b, c : a²=b²=c²=1} and H[subscript V] = {1, b} a normal subgroup of V. Since H[subscript G] and H[subscript V] are cyclic groups of order 2, set H[subscript V] = H[subscript G] = H. G and V are extensions of H by itself but are not equivalent extensions since no isomorphism exists between G and V. So the question arises: what are the necessary and sufficient conditions that two extensions G and G' of a group A by a group B be equivalent?"--Introduction. / Typescript. / "January 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaf 37).
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_257066 |
| Contributors | Lonergan, Francis Donald (authoraut), McCarthy, Paul J. (professor directing thesis.), Florida State University (degree granting institution) |
| Publisher | Florida State University, Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource (iv, 37 leaves), computer, application/pdf |
| Rights | This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them. |
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