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Tag Archives: group theory
Intermediate Group Theory (4)
Applications We’ll use the results that we obtained in the previous two posts to obtain some very nice results about finite groups. Example 1. A finite group G of order p2 is isomorphic to either Z/p2 or (Z/p) × (Z/p). In particular, it … Continue reading
Intermediate Group Theory (3)
Automorphisms and Conjugations of G We’ve seen how groups can act on sets via bijections. If the underlying set were endowed with a group structure, we can restrict our attention to bijections which preserve the group operation. Definition. An automorphism of … Continue reading
Posted in Notes
Tagged advanced, automorphisms, conjugate, group actions, group theory, semidirect products
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Intermediate Group Theory (2)
This is a continuation from the previous post. Let G act on set X, but now we assume that both G and X are finite. Since X is a disjoint union of transitive Gsets, and each transitive Gset is isomorphic to G/H for some subgroup H ≤ G, it follows that … Continue reading
Posted in Notes
Tagged advanced, cauchy's theorem, group actions, group theory, normaliser, sylow theorems
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Intermediate Group Theory (0)
Let’s take stock of what we know about group theory so far in the first series. We defined a group, which is a set endowed with a binary operation satisfying 3 properties. For each group, we considered subsets which could … Continue reading
Casual Introduction to Group Theory (6)
Homomorphisms [ This post roughly corresponds to Chapter VI of the old blog. ] For sets, one considers functions f : S → T between them. For groups, one would like to consider only actions which respect the group operation. Definition. Let G and … Continue reading
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Tagged advanced, factor through, group theory, homomorphism, isomorphism theorems, normal subgroups, universal properties
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Casual Introduction to Group Theory (5)
Normal Subgroups and Group Quotients [ This corresponds to approximately chapter V of the old blog. ] We’ve already seen that if H ≤ G is a subgroup, then G is a disjoint union of (left) cosets of H in G. We’d like to use the set … Continue reading
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Tagged advanced, group quotients, group theory, groups, normal subgroups, subgroups
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Casual Introduction to Group Theory (4)
Cosets and Lagrange’s Theorem [ This post approximately corresponds to chapter IV from the old group theory blog. ] The main theorem in this post is Lagrange’s theorem: if H ≤ G is a subgroup then H divides G. But first, let’s consider … Continue reading
Posted in Notes
Tagged advanced, cosets, double cosets, group theory, groups, lagrange's theorem
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