aaaaaaaa a ABEL1AN account all all an and and and and and and and and and

and,

being bounded bounded by by by by

cannot case class. clear clearly conclude contain contains contrary contrary copies cyclic

deduce direct direct direct divisible divisible divisible divisible,

each each exact exact Example $\mathrm{E}\mathrm{X}\mathrm{T}\mathrm{\sim }\mathrm{R}\mathrm{E}\mathrm{P}\mathrm{R}\mathrm{O}\mathrm{D}\mathrm{U}\mathrm{C}\mathrm{E}\mathrm{D}$

follows follows, for for for for for for free from

group group groups groups, GR0UPS.

has have have Hence Hence Hence Hence Hence hence, hoids

If if implies implies implies infinite infinite infinite infinite infinite. integer. is is is is is

is is is is is is is is is it it it it

maximal maximal, Moreover, Moreover, must

$\mathit{H}\mathrm{\cong }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{H}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathit{W}\mathrm{/}\mathit{t}\mathit{W}\mathrm{\cong }\mathit{Q}\mathrm{\left(}\mathfrak{n}\mathrm{\right)}\mathrm{,}$ $\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{Q}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{\cong }\mathit{Q}$. $\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{Q}\mathrm{,}\mathit{X}\mathrm{)}\mathrm{\ne }\mathrm{0}\mathrm{.}\mathrm{0}\mathrm{\to }\mathit{t}\mathit{W}\mathrm{\to }\mathit{W}\mathrm{\to }{\mathit{Q}}^{\mathrm{\left(}\mathfrak{n}\mathrm{\right)}}\mathrm{\to }\mathrm{0}$

! $\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{G}\mathrm{/}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{=}\mathrm{0}\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{G}\mathrm{/}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{\cong }{\mathit{Q}}^{\mathrm{\prime }\mathrm{7}\mathit{L}}$,

$\mathit{H}\mathit{H}\mathfrak{M}\mathfrak{M}\mathit{m}\mathfrak{n}\mathit{p}\mathit{Q}\mathit{G}\mathrm{,}$ $\mathfrak{M}\mathrm{,}$ $\mathit{p}\mathrm{,}$ $\mathit{p}$. $\mathit{t}\mathit{G}\mathit{t}\mathit{G}$ tH $\mathit{H}\mathrm{\in }\mathfrak{M}\mathit{G}\mathrm{\in }\mathfrak{M}$. $\mathit{H}\mathrm{\cong }{\mathit{Q}}^{\mathrm{;}\mathrm{?}}$. $\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{p}}\mathrm{\right(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{\rho }}\mathrm{(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{\rho }}$

$\mathrm{f}\mathit{H}\mathrm{\ne }\mathrm{0}\mathrm{,}$ $\mathit{X}\mathrm{/}\mathit{p}\mathit{X}\mathrm{\ne }\mathrm{0}\mathrm{(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{\rho }}\mathrm{\ne }\mathrm{0}\mathrm{(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{\rho }}\mathrm{\ne }\mathrm{0}\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{Q}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{W}\mathrm{,}\mathrm{\to }\mathit{X}\mathrm{)}$,

$\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\right(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{p}}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\otimes }\mathrm{(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{)}\mathrm{)}\mathrm{\ne }\mathrm{0}\mathrm{0}\mathrm{\to }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}{\mathit{Q}}^{\mathrm{\left(}\mathfrak{n}\mathrm{\right)}}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{\to }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{W}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\left)}\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\right(}{\mathit{Q}}^{\mathrm{\left(}\mathfrak{n}\mathrm{\right)}}\mathrm{,}\mathrm{}\mathit{X}\mathrm{.}\mathrm{)}\mathrm{\cong }\mathrm{(}\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{Q}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}{\mathrm{)}}^{\mathfrak{n}}$

$\mathit{G}\mathrm{\cong }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{G}\mathrm{/}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{\oplus }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}$. $\mathit{t}\mathit{W}\mathrm{\cong }{{\displaystyle ⨁}}_{\mathit{p}\mathrm{\in }\mathit{P}}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\right(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathit{p}}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\otimes }\mathrm{(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{)}\mathrm{)}$

$\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{(}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{)}\mathrm{\cong }\mathrm{|}{\mathrm{I}}_{\mathrm{-}}^{\mathrm{-}}{\mathrm{I}}^{\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\right(}\mathit{t}\mathit{G}{\mathrm{)}}_{\mathrm{/}}\mathrm{,}\mathrm{,}\mathit{X}}\mathrm{\otimes }\mathrm{(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{\left)}\mathrm{\right)}\mathrm{=}\mathrm{t}\mathit{V}$

p@P

$\mathit{\alpha }\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{\text{-}}$ non-zero not Now number number

of of of of of of of of of of of of on only only

$\mathrm{\left|}\mathrm{8}\mathrm{1}\mathrm{2}\mathrm{4}\mathrm{5}\mathrm{\right)}\mathrm{,}$ $\mathrm{2}\mathrm{.}\mathrm{3}$ finite finite, finite. finite. ([2], ([2], ([5], (b) (iii), (ii), (ii), (ii). (ii). (i), (iv),

(iv).

positive $\mathrm{p}$. p. 136), p. 44). primes primes primes primes. proof proves

rank, rank, rank, reducedness relevant return rily

$\stackrel{\mathrm{`}}{\mathrm{3}}\mathrm{a}\mathrm{y}\mathrm{,}$ sequence sequence since since since, subgroup subgroup such sum sum summand

that that that that that that that that that that The The the the the the then then

rherefore This this this to to to torsion-free torsion-free torsion-free

We We we we we were where where which

yields