  We consider amalgamation properties of convergent sequences in topological
groups and topological vector spaces. The main result of this paper is that,
for arbitrary topological groups, Nyikos's property $\alpha_{1.5}$ is
equivalent to Arhangel'ski\u{\i}'s formally stronger property $\alpha_1$. This
result solves a problem of Shakhmatov (2002), and its proof uses a new
perturbation argument. We also prove that there is a topological space $X$ such
that the space $C_p(X)$ of continuous real-valued functions on $X$, with the
topology of pointwise convergence, has Arhangel'ski\u{\i}'s property $\alpha_1$
but is not countably tight. This result follows from results of
Arhangel'ski\u{\i}--Pytkeev, Moore and Todor\v{c}evi\'c, and provides a new
solution, with remarkable properties, to a problem of Averbukh and Smolyanov
(1968) concerning topological vector spaces. The Averbukh--Smolyanov problem
was first solved by Plichko (2009), using Banach spaces with weaker locally
convex topologies.
