The Fundamental Theorem of Cyclic Groups (FTCG) is very helpful to construct the subgroup lattice for a cyclic group. This is because there is exactly one subgroup of each order that divides the order of the whole group. For the cyclic group of order 20, ℤ20 = ＜1＞={0,1,2,3,...,19} under addition modulo 20, the unique subgroup of order 10 is ＜2＞={0,2,4,6,8,10,12,14,16,18}, the unique subgroup of order 5 is ＜4＞={0,4,8,12,16}, the unique subgroup of order 4 is ＜5＞={0,5,10,15}, the unique subgroup of order 2 is ＜10＞={0,10}, and the unique subgroup of order 1 is the trivial subgroup ＜0＞={0}. We can form a directed graph from these subgroups using set inclusion. This is the subgroup lattice. It's pretty tricky and easy to make a mistake, so be careful! • Abstract Algebra Cours... .
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29 фев 2024
