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Galois Theory, 4th Edition
book

Galois Theory, 4th Edition

by Ian Nicholas Stewart
March 2015
Intermediate to advanced content levelIntermediate to advanced
344 pages
10h 18m
English
Chapman and Hall/CRC
Content preview from Galois Theory, 4th Edition
Speculations about Proofs 305
H has q conjugates, which intersect pairwise in the identity, and K has p conjugates,
which intersect pairwise in the identity. Therefore G has 1 element of order 1, at least
(p 1)q elements of order p, and at least p(q 1) elements of order q. These total
2pq p q + 1 = pq + (p 1)(q 1) elements, a contradiction since p, q > 1.
(3) Suppose G is simple of order 2p
k
. There is no subgroup of index 2, so every
proper subgroup has index divisible by p, contrary to Lemma 25.4(3).
(4) Suppose G is simple of order 3p
k
. Since 3p
k
8, Lemma 25.5 implies that there
is no subgroup of index 3. Therefore every proper subgroup
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Publisher Resources

ISBN: 9781482245837