Abelian $$p$$-groups with a fixed elementary subgroup or with a fixed elementary quotient

Abstract In his 1934 paper, G. Birkhoff poses the problem of classifying pairs ( G , U ) where G is an abelian group and $$U\subset G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> a subgroup, up to automorphisms of G . In general, Birkhoff’s problem is not considered feasible. In this note, we fix a prime number p and assume that G is a direct sum of cyclic p -groups and $$U\subset G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> is a subgroup. Under the assumption that the factor group G / U is an elementary abelian p -group, we show that the pair ( G , U ) always has a direct sum decomposition into pairs of type $$({\mathbb {Z}}/(p^n),{\mathbb {Z}}/(p^n))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> or $$(\mathbb {Z}/(p^n), (p))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Surprisingly, in the dual situation, we need an additional condition. If we assume that U itself is an elementary subgroup of G , then we show that the pair ( G , U ) has a direct sum decomposition into pairs of type $$({\mathbb {Z}}/(p^n),0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> or $$(\mathbb {Z}/(p^n), (p^{n-1}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> if and only if G / U is a direct sum of cyclic p -groups. We generalize the above results to modules over commutative discrete valuation rings.