Modules

Which of these $\mathbb{Z}$-modules are free? (Choose all that apply.)

How many abelian groups of order $72$ are there, up to isomorphism?

Which group is $\mathbb{Z}^2 / \langle (4, 6), (6, 4) \rangle$ (the quotient by the subgroup generated by these two vectors)?

A linear map $T$ on $\mathbb{C}^6$ has characteristic polynomial $(x - 2)^6$, minimal polynomial $(x - 2)^3$, and $\dim \ker(T - 2) = 3$. What are the sizes of its Jordan blocks?

In the proof that a submodule $N \subseteq R^n$ over a PID is free, why does the sequence $0 \to N \cap R^{n-1} \to N \to \pi(N) \to 0$ split?

Over which of these rings $R$ does some submodule of a free $R$-module (of finite rank) fail to be free? (Choose all that apply.)

How many elements does $\mathbb{Z}/12 \otimes_{\mathbb{Z}} \mathbb{Z}/18$ have?

Which Lean statement expresses that tensoring with $\mathbb{Z}/2$ does not preserve injectivity (as proved in tensor_not_left_exact)?

Write invariant_factors(orders): given positive integers $n_1, \dots, n_k$, return the invariant factors of $\mathbb{Z}/n_1 \oplus \cdots \oplus \mathbb{Z}/n_k$ as a list $[d_1, d_2, \dots]$ with $1 < d_1 \mid d_2 \mid \cdots$ (empty for the zero group). For example $\mathbb{Z}/4 \oplus \mathbb{Z}/6 \cong \mathbb{Z}/2 \oplus \mathbb{Z}/12$, so invariant_factors([4, 6]) == [2, 12].