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H(n + m) = xn+m = xn xm = h(n)h(m) , thus h is a homomorphism. If h(n) = xn = xm = h(m) , then n = m . thus h is injective. For every xn ∈ G , the integer n goes to xn under h . Hence, h is surjective. Example. Consider the infinite cyclic subgroup of (R∗ , ·) (the nonzero reals with the usual multiplication) generated by the element x= √ 12 2 (1) The number x is, by definition, the quotient of the frequencies between a pitch and another that is a semitone above (see chapter 4, section 1). The numbers xk (up to multplication by a constant) correspond to the hertzian frequencies of the pitches used in the music of equal tempered tuning.

For q = 6 , m = 2 , we get {11·6 , 12·6 = 112 = 0} which, in additive notation, is {6 · 1, 12 · 1 = 0} = {6, 0} = (6) . Finally, for q = 12 , m = 1 , we get {11·12 = 0} which, in additive notation, is {12 · 1 = 0} = {0} = (0) = O . Thus we have a subgroup diagram of Z12 : ! 1 Let h : G −→ G be a homomorphism of multiplicative groups. Show that h(xn ) = (h(x))n , n∈Z. 2 Show that the multiples of Z , nZ with n ∈ Z , are subgroups in Z . 3 Show that every subgroup of Z es cyclic. 4 Show that any subgroup of a cyclic group is cyclic.

As the sequence is exact, H = im f = ker g and im g = ker h = e ; then, g is the trivial homomorphism. Inversely, if g is the trivial homomorphism, then f is an epimorphism and h is a monomorphism. 6 Proposition. If f g h H −→ H −→ G −→ G is an exact sequence of groups, h is monomorphism if and only if g is trivial; g is trivial if and only if f is an epimorphism. So, when we have a short exact sequence as follows f g e −→ G −→ G −→ G −→ e we will write it, indistinctly, as G where denotes injective and f G g G surjective.

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