Is there a name for the property of a permutation* group of a set S*, $G$, such that:
For at least one permutation $\alpha \in G$ and at least one $i$ of S*: $\alpha(i)\neq i$, but the transposition $(\alpha(i) \ i)\notin G$.
(Cyclic groups like $\mathbb{Z}_4$ have this property, and so does e.g. $D_4$.)
Or a name for this property?
For all $\alpha \in G$, if $\alpha(i)\neq i$, then $(\alpha(i) \ i) \in G$ and at least one $\alpha \in G, \alpha(i)\neq i$?
Or, names for properties equivalent to these?
(*) edited
[Edited the infinite case]
Your two properties are the negation of each other, and the second one is the rare one – though not quite as rare as Derek Holt claims. I don't really think there's a name for this, but we can at least classify the such finite permutation groups pretty easily.
Assume $G$ is finite and has the second property. Let $\mathcal O(a)=\{\alpha(a)\mid\alpha\in G\}$ be the orbit of $a$. Then $G$ contains all transpositions of elements in $\mathcal O(a)$, since if $b,c\in \mathcal O(a)$, say $b=\alpha(a)$ and $c=\beta(a)$, then $$ b=\alpha(\beta^{-1}(c)) \implies (b\ c)\in G. $$ But that means $G$ has a subgroup acting as the symmetric group on $\mathcal O(a)$.
So let $\mathcal O_1,\ldots,\mathcal O_k$ be the orbits of $G$ (assume for now there are finitely many orbits). Then $G$ is isomorphic to a direct product of symmetric groups, $$ G \cong S_{|\mathcal O_1|} \times \cdots \times S_{|\mathcal O_k|}. $$ To prove this, we need to show that the symmetric subgroups are normal, essentially disjoint, and generate $G$ – which is all easy.
If $G$ is infinite there is more freedom. If we assume all the orbits $\mathcal O_i$, $i\in I$ are finite, then we can say that $$ \bigoplus_{i\in I} Sym(\mathcal O_i) \le G\le \prod_{i\in I} Sym(\mathcal O_i), $$ where the above are a direct sum and a direct product. If there are orbits of infinite cardinality, then the transpositions don't generate symmetric groups, so the situation is even more open. I think we then can say $$ \bigoplus_{i\in I} FSym(\mathcal O_i) \le G\le \prod_{i\in I} Sym(\mathcal O_i), $$ where $FSym(S)$ is the finitary symmetric group on the set $S$. But I haven't thought it all the way through.