F_p vector space is p-torsion abelian group

Proposition

Let 20260202-f_p_vector_space_is_p_torsion_abelian_group_82d648c34592a2b53436a392fc3bcb25bfa5e40a.svg be a prime number and an abelian group.
Suppose for each

20260202-f_p_vector_space_is_p_torsion_abelian_group_a1fa65f3a08cb7cf43fa7a729146df79f9d0e07d.svg

Then 20260202-f_p_vector_space_is_p_torsion_abelian_group_bc693172e3e84a891cfc37b25c000aa67abdac66.svg is the underlying additive group of a vector space.

Proof

Let 20260202-f_p_vector_space_is_p_torsion_abelian_group_aa0210698c870c42c0806f295528916f1e1d5d65.svg and .
Then define .

This definition is welldefined, since for 20260202-f_p_vector_space_is_p_torsion_abelian_group_a41eb27548d09a98574284f54a32109cf33e31d8.svg

20260202-f_p_vector_space_is_p_torsion_abelian_group_71093f7ef4692dfe214190aa6e37e25fdfe60287.svg

Then one can check, that this multiplication give sa

Remark

this is a proof using methods from commutative algebra:

Let 20260202-f_p_vector_space_is_p_torsion_abelian_group_fceaf1888cb184a11f8f0a24003ac2da3d6a9639.svg be an abelian group, the ring consisting of group-isomorphism (under pointwise addition and composition).

Then a 20260202-f_p_vector_space_is_p_torsion_abelian_group_b38a59450bab55982b006cf9aa75bfd47f38ce85.svg vector space structure is precisely a ring homomorphism .
If every element in is -torsion, this amounts precisely to the statement that the map

20260202-f_p_vector_space_is_p_torsion_abelian_group_5414762badf0c6a1f7feff3d891deffa11d4eecc.svg

is the zero morphism.
So by the universal property of 20260202-f_p_vector_space_is_p_torsion_abelian_group_c8a26f3714e77a3d68d7258be639c1967df362ff.svg there exists an extension

20260202-f_p_vector_space_is_p_torsion_abelian_group_1650ebf149dfb32c30166a4038c20ee3a7eb6e23.svg

Fp vector space is p-torsion abelian group

Proposition

Proof

Remark

F_p vector space is p-torsion abelian group