We define the notion of a subgroup of a group. Then we cover several subgroup tests and demonstrate them with examples. Next, we define cosets and show basic properties of cosets with a couple of elementary examples. Finally, we prove that all cosets of a particular subgroup have the same size, the number of left and the number of right cosets of a subgroup match, and establish Lagrange's theorem. A quick corollary shows prime implies cyclic.
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