Abelian Groups - László Fuchs [Springer]

1 Basic Definitions 5
A subgroup E of A is essential if its intersection with any non-zero subgroup of
A is non-zero. It is easily checked that (i) E is essential in A if and only if it contains
the socle of A and A=E is torsion; (ii) the property of being an essential subgroup is
transitive; and (iii) the intersection of two essential subgroups is again an essential
subgroup.
F Notes. Theorem 1.2 is an elementary, but fundamental result. It means that a typical group
can be thought of as being a composite of a torsion and a torsion-free group. This, however, does
not reduce the theory of mixed groups to those of these constituents, since a major issue that
remains is to find out how they are glued together to form the mixed groups. It is hard to trace the
history of Theorem 1.2.
Generalizations of ‘torsion’ exist for modules, albeit not over all rings. If we mean by a ‘torsion’
element one whose annihilator in the ring is ¤ 0, then the left Ore domains are exactly those rings
R for which in every left R-module M the torsion elements form a submodule T and M=T has
no torsion ¤ 0. There is an extensive literature on torsion theories in module categories, even in
additive categories; see, e.g., J. Golan’s book Torsion Theories (1986).
Exercises
(1) The associativity and commutativity laws can be combined into a single law:
.a C b/ C c D a C .c C b/ for all a; b; c 2 A.
(2) (a) Let B 1 ;:::;B k be subgroups of the group A, andletB D B 1 \\B k .
The index jA W Bj is not larger than the product of the indices jA W B i j.
(b) The intersection of a finite number of subgroups of finite index is again a
subgroup of finite index.
(3) Let B; C be subgroups of A.
(a) For every a 2 A, thecosetsa C B and a C .B C C/ have non-zero
intersections with the same cosets mod C.
(b) A coset mod B contains exactly jB W .B \ C/j pairwise incongruent
elements mod C.
(4) (O. Ore) The group A has a common system of representatives mod two of its
subgroups, B and C, if and only if jB W .B \ C/j DjC W .B \ C/j. [Hint: for
necessity use Exercise 3; for sufficiency, divide the cosets mod B into blocks
mod .B C C/, and define a bijective correspondence within the blocks.]
(5) (N.H. McCoy) (a) Let B; C; G be subgroups of A such that G is contained in
the set union B [ C. Then either G B or G C. [Hint:ifb 2 .B \ G/ n C,
then c 2 C \ G implies b C c 2 B \ G; c 2 B \ G.]
(b) The same fails for the set union of three subgroups.
(6) If n D p i 1
1
p i k
k
is the canonical representation of the integer n >0,then
nA D p i 1
1
A \\p i k
k
A and AŒn D AŒp i 1
1
˚˚AŒp i k
k
.
(7) (Honda) If B A and m 2 N, setm 1 B Dfa 2 A j ma 2 Bg. Prove that
(a) m 1 B is a subgroup of A containing B; (b)m 1 0 D AŒm; (c)m 1 mB D
B C AŒm; (d)m.m 1 B/ D B \ mA; (e)m 1 n 1 B D .mn/ 1 B where n 2 N.