By Thomas Keilen

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The following theorem is stated for right cosets, but it is equally true for left cosets. 4 (1) If a ∈ F, then Fa = F. (2) Two right cosets of F are either identical or they are disjoint. Proof: (1) If a ∈ F, then certainly Fa ⊆ F. It is possible to prove the much stronger statement, however, that Fa contains every element of F once and only once. Let b be an arbitrary element of F, and deﬁne c = ba−1 ∈ F. Then ca ∈ Fa and ca = b, so Fa contains every element of F at least once. Now suppose that it contained an element more than once.

The region labeled U i+1 is similarly obtained from ai+1 ai · · · a1 e. Any point 60 CHAPTER 3. FORMAL PROPERTIES OF LIE GROUPS c in the overlap region has two sets of parameters, β and β ′ ; consequently there is a mapping or function that expresses β in terms of β ′ and vice versa. A set of elements like U i that can all be speciﬁed with a single system of parameters (ie. without reparameterizing) is called a coordinate system patch. By chosing diﬀerent elements to play the role of c and diﬀerent chains of ai ’s we can cover the entire group G0 with interlocking coordinate patches.

The set of all 2 × 2 special orthogonal matrices is a one-parameter group that can be parameterized as follows. ( M (α1 ) = cos α1 sin α1 − sin α1 cos α1 ) The curve generated by α1 = 2πt with 0 ≤ t ≤ 1 starts at M (0) = I and ends at the same point. In this case the ends points are identical, and the curve is a loop. Another path between I and I is generated by α1 = 0t. These paths cannot be continuously deformed into one another, since only α1 = 2πn with n equal to an integer can produce M (α1 ) = I.

### Algebra I [Lecture notes] by Thomas Keilen

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