Page 139 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Self Assessment

1. A group G is a p-group of every element in G has its order a power of ................
(a) g (b) p

(c) g-1 (d) p-1
2. If G is a finite group. Then G is p-group of and only if |G| = ................
(a) p -p (b) p p

(c) p n (d) p n
3. Let P be a Sylow p-subgroups of a ................ G and let x have as its order a power of p. If
x p(x) = p. Then x  p.
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(a) indirect (b) infinite
(c) finite (d) direct
4. A subgroup of a group G is a p- ................ if it is a p-group.
(a) subgroup (b) normal group
(c) infinite group (d) cyclic group

5. How many elements of order 7 are there is a simple group of order 168.
(a) 7 (b) 8
(c) 9 (d) 48

12.2 Summary

Let G be a finite group and p a prime such that p divides the order of G. Then G contains a

subgroup of order p.
(First Sylow Theorem) Let G be a finite group and p a prime such that p divides |G|.
r

Then G contains a subgroup of order p . r
Let P be a Sylow p-subgroup of a finite group G and let x have as its order a power of p.

If x Px = P. Then x  P.
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Let H and K be subgroups of G. The number of distinct H-conjugates of K is

[H : N(K)  H].
(Second Sylow Theorem) Let G be a finite group and p a prime dividing |G|. Then all

Sylow p-subgroups of G are conjugate. That is, if P and P are two Sylow p-subgroups,
2
1
there exists a g  G such that gP g = P .
-1
1
2
12.3 Keywords
Cauchy: Let G be a finite group and p a prime such that p divides the order of G. Then G contains
a subgroup of order p.

First Sylow Theorem: Let G be a finite group and p a prime such that p divides |G|. Then G
r
contains a subgroup of order p .
r

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