The Theory of p-Groups [Lecture notes] by David A. Craven

By David A. Craven

Show description

Read Online or Download The Theory of p-Groups [Lecture notes] PDF

Similar linear books

Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics

This ebook is meant as an introductory textual content with reference to Lie teams and algebras and their function in a variety of fields of arithmetic and physics. it's written by way of and for researchers who're essentially analysts or physicists, no longer algebraists or geometers. no longer that we have got eschewed the algebraic and geo­ metric advancements.

Dimensional Analysis. Practical Guides in Chemical Engineering

Functional publications in Chemical Engineering are a cluster of brief texts that every offers a concentrated introductory view on a unmarried topic. the entire library spans the most subject matters within the chemical strategy industries that engineering execs require a uncomplicated realizing of. they're 'pocket guides' that the pro engineer can simply hold with them or entry electronically whereas operating.

Linear algebra Problem Book

Can one research linear algebra completely via fixing difficulties? Paul Halmos thinks so, and you'll too when you learn this publication. The Linear Algebra challenge publication is a perfect textual content for a direction in linear algebra. It takes the scholar step-by-step from the elemental axioms of a box throughout the inspiration of vector areas, directly to complex options equivalent to internal product areas and normality.

Additional resources for The Theory of p-Groups [Lecture notes]

Example text

If A stabilizes some normal series, then A = 1. Proof: Let 1 = H0 ··· H1 Hr = G be a series, and suppose that φ ∈ A stabilizes this series. Then H1 that if Hi if Hi CG (φ), then Hi+1 CG (φ). We will show CG (φ), proving the result, since if G = CG (φ), φ = 1. 2) Hi+1 /Hi = CHi+1 /Hi (φ) = CHi+1 (φ)Hi /Hi = CHi+1 (φ)/Hi , since Hi CG (φ) ∩ Hi+1 . Thus Hi+1 = CHi+1 (φ) CG (φ), as required. 5 (Thompson’s Critical Subgroup Theorem) Let G be a finite p-group. Then there is a characteristic subgroup C of G such that: (i) every non-trivial p -automorphism of G induces a non-trivial p -automorphism on C; (ii) CG (C) = Z (C); (iii) [G, C] Z (C) (or equivalently [G, C, C] = 1); and (iv) C has class at most 2, and C/ Z (C) is elementary abelian.

If X is cyclic, then G contains a cyclic subgroup of index 2, and preimage of X we easily see that G has maximal class, so we assume that X = x, z is isomorphic with n−3 C2n−2 × C2 . As with the previous theorem, we get x2 four-group, generated by x 2n−3 ∈ Z (G), and so Z (G) is a Klein = y and z. Now consider H = G/ y . This is again of maximal class, and the image of X in H is isomorphic with C2 × C2n−3 . Since no group of maximal class has a non-cyclic abelian subgroup of order 8, this forces n = 4.

Then G is an abelian group of odd order. 2. This implies that xφ = x−1 , and so this map is an automorphism. It is also an anti-automorphism, and so G possesses an automorphism that is also an anti-automorphism; thus G is abelian. 8 (B. Neumann, 1956) Let φ be a fixed-point-free automorphism of G, and suppose that φ has order 3. Then G is nilpotent of class at most 2. 2, we see that xxφ xφ = 1 for all x ∈ G. Let y = x−1 , so that 2 yφ yφy = 1 for all y ∈ G. Therefore 2 [x, xφ ] = yy φ xxφ = xφ xxφ = 1.

Download PDF sample

Rated 4.56 of 5 – based on 45 votes