Abstract Algebra Chapter II
Name: A HS W 8!"
Totally 110 points, including 10 bonus ones. Please nish in 50 minutes. Here
(3) Sn denotes the symmetric group of n letters;
(b) The order of a finite gro
1.6. CYCLIC SUBGROUPS
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1.6
Cyclic Subgroups
Recall: cyclic subgroup, cyclic group, generator. Def 1.68. Let G be a group and a G. If the cyclic subgroup a is finite, then the order of a is | a |. Ot
2.3. COSETS AND THE THEOREM OF LAGRANGE
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2.3
Cosets and the Theorem of Lagrange
We always assume that H is a subgroup of the group G.
2.3.1
Cosets
Def 2.33. Let H be a subgroup of G. Given a G, the
3.4. (III-16) GROUP ACTION ON A SET
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3.4
3.4.1
(III-16) Group Action on a Set
Group Action
We have seen many examples of group acting on a set. Ex 3.54. The group D4 of symmetries of a square. Ex 3.
Chapter 4
7. Advanced Group Theory
It is important to build up the correct visions about things in a group, a homomorphism, or so.
4.1
VII-34. Isomorphism Theory
Thm 4.1 (First Isomorphism Theorem). L
4.2. VII-35. SERIES OF GROUPS
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4.2
VII-35. Series of Groups
To give insights into the structure of a group G, we study a series of embedding subgroups of G.
4.2.1
Subnormal and Normal Series
Def 4.1
4.3. 36. SYLOW THEOREMS AND APPLICATIONS
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4.3
36. Sylow Theorems and Applications
The structures of finite abelian groups are well classified. The structures of finite nonabelian groups are much mor
01.08 (1 + i)3 = 1 + 3i + 3i2 + i3 = 1 + 3i - 3 - i = -2 + 2i 01.17 z 4 = -1 = ei . So z = ei(/4+2k/4) for k = 0, 1, 2, 3. Note that ei = cos + i sin 01.38 eia eib = ei(a+b) . By Euler's formula, (cos