Abstract
Consider a finite group G together with a subgroup H of G. Are the orders of H and G related in any way? Assuming H is not all of G, choose an element g 1 from G — H, and multiply every element of H on the left by g 1 to form the set
We claim that g 1 H has the same size as H and is disjoint from H. The first assertion follows because the correspondence h → g 1 h from H to g 1 H can be inverted (just multiply every element of g 1 H on the left by \(g_1^{ - 1}\)) and is therefore a bijection. For the second, suppose x lies in both H and g 1 H. Then there is an element h 1 ∈ H such that x = g 1 h 1. But this gives \({g_1} = xh_1^{ - 1}\), which contradicts our initial choice of g 1 outside H.
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© 1988 Springer Science+Business Media New York
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Armstrong, M.A. (1988). Lagrange’s Theorem. In: Groups and Symmetry. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4034-9_11
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DOI: https://doi.org/10.1007/978-1-4757-4034-9_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3085-9
Online ISBN: 978-1-4757-4034-9
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