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Example 32.1 (unitarymatrices) ta g = un ⊂ gln(c) For example, we do not view (q∗, ×) as being contained in (r, +) because the operations are diferent E ∈ un t ∈ r let last time,weshowedthattheconditionthat for all is equivalentto a∗ = −aa a requiring that

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`+1(g) ≤ ep`+1+1 ≤ dp`+1+1 It is important not only that one set is contained in the other, but the operation must also be the same Thus power subgroups behave better than omega subgroups, a phenomenon that one also encounters in other situations

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Subgroups, cyclic groups and generators 2.1 de nition

A subgroup of a group g is a subset h g which is also a group using the same operation as in g When h is a subgroup of g, we write h g. Let p be a prime In particular, if ω generates ω1 and e has a canonical subgroup ce,

We need to nd a smooth homomorophism from r to g using lx This homomorphism is provided by the ow or integral curve of lx But let us work this out in more. For each integer , let p e se( ) be the set of subgroups of g e order and put se( ) = |se( )

Based on theorem l, frobenius proved

P theorem 2 (frobenius [fr1, section 4.i]) 1 subsets and subgroups we have seen many examples of groups contained in larger groups For example, (z, +) is contained in (q, +), which itself is contained in (r, +)