On a finiteness condition in topological groups (Q800506)

For locally compact groups G, \(G_ 0\) denotes the connected component of identity, \(B=B(G)\) the periodic part, \(S_ p(G)\) the p-Sylow subgroup, r(G) the rank of G as defined by Maltsev, \(I_ p\) (resp., \(R_ p)\) the additive group of the ring of p-adic integers (resp., of the field of p- adic numbers). A U-group is a locally nilpotent, torsion-free group. A group G is an A-group if every finite set of its subgroups of finite rank generates another such, an \(\hat A\)-group if \(r(G/\cap^{k}_{i=1}T_ i)<\infty\) whenever \(r(G/T_ i)<\infty\quad (1\leq i\leq k).\) The prinipal results are as follows. A locally nilpotent group is an \(\hat A\)-group if \(r(G/BG_ 0)<\infty\). An Abelian U-group is an \(\hat A\)-group iff it is discrete or \(r(G/BG_ 0)<\infty.\) A nilpotent topological group G is an A-group iff (1) \(r(B_ 0)<\infty\) or (2) \(r(B_ 0)=\infty,\quad G=B(G),\) and \(G/B_ 0=H\times\prod^{k}_{i=1}S_{p_ i}\), where H is compact and no \(S_{p_ i}\) contains a subgroup of form \(p_ i^{\infty}\) or \(R_{p_ i}\). An Abelian group G is an \(\hat A\)-group iff \((1)\quad r(G/G_ 0B)<\infty\), or (2) G is totally disconnected and \(B=D\times\prod^{k}_{i=1}S_{p_ i}\), where D is discrete and no \(S_{p_ i}\) contains a subgroup of type \(I_ p\) or \(R_ p\). A U- group G is an A-group iff (1) \(r(B_ 0)<\infty\), or (2) \(r(B_ 0)=\infty,\quad G=B(G)\) and \(G/B_ 0=H\times\prod^{k}_{i=1}S_{p_ i}\), where H is compact and no \(S_{p_ i}\) contains a subgroup of type \(R_{p_ i}\).