In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H. The Bender–Suzuki theorem, proved by extending work of Suzuki , classifies the groups G with a strongly embedded subgroup H. It states that either 1. * G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution 2. * or G/O(G) has a normal subgroup of odd index isomorphic to one of the simple groups PSL2(q), Sz(q) or PSU3(q) where q≥4 is a power of 2 and H is O(G)NG(S) for some Sylow 2-subgroup S. , part II) revised Suzuki's part of the proof. extended Bender's classification to groups with a proper 2-generated core. (Wikipedia).
In this tutorial we define a subgroup and prove two theorem that help us identify a subgroup. These proofs are simple to understand. There are also two examples of subgroups.
From playlist Abstract algebra
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From playlist Abstract Algebra
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Center of a group in abstract algebra
After the previous video where we saw that two of the elements in the dihedral group in six elements commute with all the elements in the group, we finally get to define the center of a group. The center of a group is a subgroup and in this video we also go through the proof to show this.
From playlist Abstract algebra
Abstract Algebra | The notion of a subgroup.
We present the definition of a subgroup and give some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
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From playlist ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (2022)
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From playlist Abstract Algebra