Mathematical Methods of Theoretical Physics by Karl Svozil

Author:Karl Svozil
Language: eng
Format: epub
Publisher: World Scientific Publishing Co. Pte. Ltd.
Published: 2020-07-15T00:00:00+00:00

and thus we obtain g2 H ⊂ g1 H. A similar, symmetric argument yields g1 H ⊂ g2 H; therefore, g2 H = g1 H. That is, stated pointedly, if the two sets g1 H and g2 H are not disjoint they must be identical. In the first case of identical sets g1 H = g2 H, H = (g1)–1 g2 H, and thus, by the rearrangement theorem (cf. Section 3.1.6, page 176), (g1)–1 g2 ∈ H. At the same time, if one considers all g ∈ G, and forms g H, already the elements g I = g recovers the entire group G. (Note that I ∈ H.)

For any finite group G and any subgroup H ⊂ G, the relation x ~ y : x H = y H⇔ x–1y ∈ H defines an equivalence relation. on G. Thereby, the set x H with x ∈ G is a left coset of H in G with respect to g. A similar statement applies to right cosets.

This result is part of Lagrange’s theorem in the mathematics of group theory.

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