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Abstract Algebra

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Title: Abstract Algebra

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See the first part of this post here. Throughout, denote, respectively, the centralizer and the normalizer of a subgroup of a group Also, denotes the Euler’s totient function. In the first part of this post, we defined rational groups. We showed in Proposition 1, ii), that a finite group is rational if and only if … …


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Throughout this post, is a finite group and denote, respectively, the centralizer and the normalizer of a subgroup in Also, denotes the Euler’s totient function and denotes the order of By the Lemma in this post, is isomorphic to a subgroup of for any subgroup of If is isomorphic to the whole group for any […]


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See the first two parts of this post here and here. Let be, respectively, the symmetric group and the alternating group on In this post, we find the commutator subgroups of Proposition 1. Proof. Since is abelian, Suppose now that Since is abelian, To show that recall that is generated by -cycles, and so we […]


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By the second part of the Proposition in the first part of this post, if and only if is abelian. The opposite situation is when Definition. We call a group perfect if Example 1. Every characteristically simple non-abelian group is perfect. Proof. By the sixth part of the Proposition in the first part of this […]


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Throughout this post, is a group. The Subgroup Generated By A Set. Let be a non-empty subset of Recall that the subgroup generated by is defined to be the smallest subgroup of that contains i.e., is the intersection of all the subgroups of that contain More explicitly, is the set of all the elements of […]


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