# Tag Info

Accepted

### $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Let $A$ be the additive group of bounded sequences of elements of $\mathbb{Z}[\sqrt{2}]$. Clearly $A\cong A\oplus\mathbb{Z}[\sqrt{2}]\cong A\oplus\mathbb{Z}^2$ as abelian groups, so we just need to ...

### Results from abstract algebra which look wrong (but are true)

The free group with infinitely many generators is a subgroup of the free group with two generators.

### Results from abstract algebra which look wrong (but are true)

Let $G$ be a finite group and $n \mid |G|$. If $S = \{x \in G : x^n = 1\}$ contains exactly $n$ elements, then $S$ is a subgroup of $G$. There seems no a priori reason to expect $S$ to be a subgroup ...

### Results from abstract algebra which look wrong (but are true)

Every finite simple group can be generated by at most $2$ elements. This is another famous consequence of the classification.
Accepted

### Constructively, is the unit of the “free abelian group” monad on sets injective?

Yes; this is given in Mines, Richman, Ruitenburg, A course in constructive algebra, on p.54, stated just before Lemma 4.1, and proved in that lemma by a Church-Rosser argument. (Thanks to Thierry ...
Accepted

### Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?

A great survey on this and some related topics is Osofsky's "The subscript of $\aleph_n$, projective dimension, and the vanishing of $\varprojlim^{(n)}$." As far as I am aware, this 1974 ...
Accepted

### Do these properties of a countable abelian group guarantee a Prüfer subgroup?

Yes, it must. And $G$ doesn't need to be countable. Let $H$ be the $p$-primary component of the torsion subgroup of $G$. Then the natural map $H/pH\to G/pG$ is injective, so $H$ also satisfies (1), ...
### On $p$-groups with abelian automorphism group
I suggest you to have a look at the following paper and references therein: V.K. Jain, P.K. Rai, M.K. Yadav: On finite $p$-groups with abelian automorphism group. Internat. J. Algebra Comput. 23 (2013)...