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The aim of this paper is to develop the theory of groups definable in the$p$-adic field ${\mathbb Q}_p$, with ``definable $f$-generics" in the sense ofan ambient saturated elementary extension of ${\mathbb Q}_p$. We call suchgroups definable $f$-generic groups.  So, by a ``definable f-generic'' or dfg group we mean a definable group in asaturated model with a global f-generic type which is definable over a smallmodel. In the present context the group is definable over ${\mathbb Q}_p$, andthe small model will be ${\mathbb Q}_p$ itself. The notion of a dfg group isdual, or rather opposite to that of an fsg group (group with ``finitelysatisfiable generics") and is a useful tool to describe the analogue of torsionfree o-minimal groups in the $p$-adic context.  In the current paper our group will be definable over ${\mathbb Q}_p$ in anambient saturated elementary extension $\mathbb K$ of ${\mathbb Q}_p$, so as tomake sense of the notions of $f$-generic etc. In this paper we will show thatevery definable $f$-generic group definable in ${\mathbb Q}_p$ is virtuallyisomorphic to a finite index subgroup of a trigonalizable algebraic group over${\mathbb Q}_p$. This is analogous to the $o$-minimal context, where everyconnected torsion free group definable in $\mathbb R$ is isomorphic to atrigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show thatevery open definable $f$-generic subgroup of a definable $f$-generic group hasfinite index, and every $f$-generic type of a definable $f$-generic group isalmost periodic, which gives a positive answer to the problem raised in\cite{P-Y} of whether $f$-generic types coincide with almost periodic types inthe $p$-adic case.

On groups with definable $f$-generics definable in $p$-adically closed fields