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Groupprops β

Lattice of subgroups embeds in partition lattice

Statement

Suppose G is a group, P(G) is the partition lattice of G (i.e., the lattice whose elements are partitions of G, with \alpha \le \beta if \alpha is a finer partition than \beta. Let L(G) denote the Lattice of subgroups (?) of G.

Then, there is an injective homomorphism of lattices from L(G) to P(G), given as follows: a subgroup H of G is sent to the partition of G given by the Left coset (?)s of H in G (as per left cosets partition a group). In particular:

  1. For subgroups H, K the left coset partition arising from the Join of subgroups (?) \langle H, K \rangle is the same as the join in the partition lattice of the left coset partitions arising from H and K respectively.
  2. For subgroups H, K, the left coset partition arising from the Intersection of subgroups (?) H \cap K is the same as the meet in the partition lattice of the left coset partitions arising from H and K respectively.

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