Appendix
In this appendix a few very basic or technical things are recalled in order to make the book selfcontained in a reasonable sense.
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In this appendix a few very basic or technical things are recalled in order to make the book selfcontained in a reasonable sense.
11.1 Groups A group is a set, say G, together with a law of composition for which we shall mostly use the common symbol"." for multiplication: · : G x G -+ G: (g, g')
f---+
g · g' (or gg' for sake of simplicity).
If we use this notation, we call the group G, or, better say, the pair ( G, ·), consisting of the set and the composition, a multiplicative group. At most in the commutative case, i. e. when gg' = g' g for each g, g' E G, we use the additive notation"+" and speak of an additive group G or (G. +) in this case. Groups with a commutative composition are called abelian. The composition is assumed to be associative:
g(g'g")
= (gg')g",
for each g, g', g" E G. Moreover, we assume that there exists an element (and so G is not empty!) which is a left unit: e g = g, for every g E G. Besides this, each group element g must possess a left inverse g' with respect to e : g' g = e. It is easy to check that the left unit is also a right unit, and that it is uniquely determined. It will therefore be denoted by I or, more explicitly, by lc in the multiplicative case, by 0 or Oc in the additive notation. A left inverse is also a right inverse, and it is uniquely determined, but it, of course, depends on g. We shall indicate it by g -I in the multiplicative case, by - g in the additive notation. It is not difficult to see that the same sets turn out to be groups if we assume a right unit together with right inverses instead. In a multiplicative (additive) group we can multiply (add) also subsets: For M, N ~ G we simply put M · N := {m · n I mE M, n EN},
or, in the additive case, M + N := {m A. Kerber, Applied Finite Group Actions © Springer-Verlag Berlin Heidelberg 1999
+ n I mE
M, n EN}.
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11. Appendix
The result is called the complex product (complex sum) of these subsets. A mapping cp: G -+ H between two groups G and His called a homomorphism if the following is true: cp(g. g') = cp(g). cp(g'). Note that the dot on the left hand side means the multiplication in G while that on the right hand side indicates the multiplication in the group H. A homomorphism 8 from G into the symmetric group Sx := {n In: X-+ X, bijectively}
(the multiplication is the composition of mappings) is called a permutation representation of G on X. A homomorphism into a general linear group of a vector space V is called a linear representation of G on V. A subset H of a group (G, ·)is called a subgroup, for short: H ::; G, if and only if ( H, ·) is a group. This is equivalent to
ll.l.l
H f=- 0, and [h, h'
E
H:::} h · h'- 1
E
H].
The sets
gH := {gh I hE H} are called left cosets, and the sets
Hg:={hglhEH} are called right cosets of the subgroup H. Subgroups N ~ G for which left and right cosets are the same, g N = N g, for each g, are of particular importance, they are called normal subgroups. We abbreviate this by writing N ::::) G. Each homomorphism cp: G -+ H defines (and is defined by) a partic
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