An Introduction to the Theory of Groups |
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Page 20
... cyclic subgroup C , of order d for every d divisor d of n . Therefore , n = | G | = Σ | g ( Ca ) ] . In Exercise 2.24 , however , we saw that g ( Ca ) | = p ( d ) . □ din We can now characterize finite cyclic groups . Theorem 2.15 A group ...
... cyclic subgroup C , of order d for every d divisor d of n . Therefore , n = | G | = Σ | g ( Ca ) ] . In Exercise 2.24 , however , we saw that g ( Ca ) | = p ( d ) . □ din We can now characterize finite cyclic groups . Theorem 2.15 A group ...
Page 104
... GROUPS We now have quite a bit of information about finite abelian groups , but we still have not answered the basic question : If G and H are finite abelian groups , when are they isomorphic ? Since both G and H are direct sums of cyclic ...
... GROUPS We now have quite a bit of information about finite abelian groups , but we still have not answered the basic question : If G and H are finite abelian groups , when are they isomorphic ? Since both G and H are direct sums of cyclic ...
Page 262
... cyclic groups by a divisible group . Corollary 10.34 ( Prüfer - Baer ) † Let G be a group of bounded order , i.e. , nG = O for some integer n > 0. Then G is a direct sum of cyclic groups . Proof Let B be a basic subgroup of G. Then G ...
... cyclic groups by a divisible group . Corollary 10.34 ( Prüfer - Baer ) † Let G be a group of bounded order , i.e. , nG = O for some integer n > 0. Then G is a direct sum of cyclic groups . Proof Let B be a basic subgroup of G. Then G ...
Contents
THE ISOMORPHISM THEOREMS | 11 |
PERMUTATION GROUPS | 32 |
THE SYLOW THEOREMS | 56 |
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a₁ abelian group affine Assume automorphism B₁ commute composition series conjugacy classes conjugate Corollary cosets cyclic groups defined Definition Let denoted diagram direct product direct sum disjoint divisible example Exercise exists factor set finite group follows free abelian free group function G contains G-set G₁ G₂ group G group of order H₁ hence HINT HNN extension homomorphism implies induction infinite integer isomorphic K₁ L₁ L₂ Lemma Let F Let G Let H linear matrix nilpotent nonsingular nonzero normal series normal subgroup notation one-one correspondence p-group p-primary permutation polynomial positive words prime Proof Let relations semidirect product semigroup shows simple group solvable solvable group stable letter Steiner system subgroup H subgroup of G subgroup of order subset subword summand Sylow p-subgroup Theorem torsion-free transitive G-set transvection unique vector space W₁ word problem x₁