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Abstract - 17 19 and 23 and the centralizers of each...

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Abstract In this paper, we provide many algebraic properties of the group U(24). We start out with historical background of U(24). Next, we prove that U(24) is in fact a group. In order to do this, we first prove that U(24) is closed under multiplication mod24. Then, we prove U(24) has an identity element, which is 1. Then we prove that each element in U(24) has a unique inverse that is also in U(24). Finally, we prove U(24) is associative to satisfy the axioms of a group. We also go on to prove other algebraic properties such as the order of U(24) which is 8, and we also found all of the elements of U(24) to have order 2, except for the identity that has order 1. By the Cayley table constructed on page 1, many algebraic properties were noted such as: U(24) is Abelian, the center of U(24) is all of the elements in U(24), which are 1, 5, 7, 11, 13,
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Unformatted text preview: 17, 19, and 23, and the centralizers of each element in U(24) is all of U(24). Other findings in this paper include the fact that there is no single generator of U(24) and therefore, U(24) is not cyclic. A Lattice diagram out of the subgroups of U(24) will be provided. Auotmorphisms, homomorphisms, inner automorphisms, and isomorphisms that involve U(24) will be discussed. We find some isomorphisms of U(24) by using external and internal direct products. All of the normal subgroups of this group are listed which turn out to be all of the regular subgroups of my group. Similarly, we have factor groups, and we provide the set of right cosets of these factor groups. Finally, I give applications for U(24). For example, U(n) is used in quantum physics and can be applied to that area....
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