5. Problems

Problem O-1. In Example (5) of Figure , how many copies of $M _ 3$ can you find?

Problem O-2. In Example (7) of Figure , see how long a string of intervals you can find such that each two consecutive ones are transposes and there are no repeats.

Problem O-3. Prove that a lattice $L$ is modular if and only if transposed intervals are isomorphic under the obvious maps up and down: $[a \wedge b,a] \cong [b,a \vee b]$ under $x \mapsto x \vee b$ with inverse $x \mapsto x \wedge a$.

Problem O-4. Let $G$ be a finite abelian group. Answer the following with brief proofs.

(a) Show that if    Subgp$(G)$ has a single co-atom, then $G$ is cyclic. (Examine a group element not in the co-atom. Is the converse true?)

(b) Show that if    Subgp$(G) \cong$   n$$ then $G \cong$   Z$_ {p ^ {n-1}}$ for some prime $p$.

(c) Show that if    Subgp$(G) \cong M _ n$, the lattice of length 2 with $n$ atoms, for $n>1$, then $G \cong$   Z$_ p \times$   Z$_ p$ and $n = p+1$, or else $G \cong$   Z$_ p \times$   Z$_ q$ and $n = 2$, where $p,q$ are prime, $p \neq q$. (Recall that direct-product decompositions of $G$ into two factors correspond to pairs of complementary subgroups, a lattice-theoretic idea.)

(d) For each subgroup diagram in Figure , identify the corresponding finite abelian group (if any). You need explain only why no finite abelian group other than yours fits, not why yours does have the diagram given.

Figure: subgroup lattices

text/Bdir/abelian.eps