Subgroup lattice is sometimes said to represent the structure of a group.But does it give a complete information about the group.I mean from a given subgroup lattice is it possible to construct a new group.We can frame the question saying that whether two nonisomorphic groups can have the same lattice structure or not.Then subgroup lattice will not be sufficient to determine the structure of a group.
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5All $\mathbb Z/p$ ($p$ prime) have the same subgroup lattice, don't they? – Torsten Schoeneberg Mar 24 '20 at 03:44
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@TorstenSchoeneberg That should be an answer. – Noah Schweber Mar 24 '20 at 03:58
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1Indeed, more generally the subgroup lattice of $\Bbb Z/n\Bbb Z$ depends only upon the exponents in the prime-power factorization of $n$, not the primes themselves. – Greg Martin Mar 24 '20 at 03:59
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See also https://math.stackexchange.com/questions/2477990/how-to-prove-that-two-groups-have-the-same-lattice-of-subgroups – Gerry Myerson Mar 24 '20 at 04:56
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@NoahSchweber: Done. – Torsten Schoeneberg Mar 24 '20 at 05:28
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Yes they can; for example, all the groups $\mathbb Z/p$ for $p$ prime have the same subgroup lattice.
Note that according to Does the order, lattice of subgroups, and lattice of factor groups, uniquely determine a group up to isomorphism? (linked by Gerry Myerson in the comments) even much more refined properties do not determine a group up to isomorphism.
Torsten Schoeneberg
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