Confusion on translating k-fold transitivity of groups from Endliche Gruppen by Huppert

Question

The definition 1.7 from Endliche Gruppen, B.Huppert, vol-I, Chap.II, Pg.148 is as follows: Die Permutationsgruppe $\mathfrak G$ auf der Ziffernmenge $\Omega$ heißt $k$-fach transitiv $(k \leq |\Omega|)$, falls zu jedem Paar von geordneten Teilmengen $(a_1, \dotsc, a_k)$ und $(b_1, \dotsc, b_k)$ von $\Omega$ ein $G$ aus $\mathfrak G$ existiert mit $a_i^G = b_i$ für $i = 1, \dotsc, k$.

I was confused with the part "geordneten Teilmengen" which obviously cannot mean all ordered $k$-tuples.

In a later definition 1.10, Pg.149 the notation $\Omega^{[k]}$ is introduced: Sei $\mathfrak G$ eine Permutationsgruppe auf $\Omega$. Für jede natürliche Zahl $k$ sei $\Omega^{[k]}$ die Menge der geordneten Teilmengen der Mächtigkeit $k$ von $\Omega$. Die Festsetzung $$(a_1, \dotsc, a_k)^G = (a_1^G, \dotsc, a_k^G)$$ macht die Gruppe $\mathfrak G$ zu einer zu $\mathfrak G$ isomorphen Permutationsgruppe ${\mathfrak G}^{[k]}$ auf $\Omega^{[k]}$. Offenbar ist ${\mathfrak G}^{[k]}$ genau dann transitiv auf $\Omega^{[k]}$, wenn $\mathfrak G$ $k$-fach transitiv ist auf $\Omega$.

Now my confusion is : $\Omega^{[k]}$ seems to be the set of all $k$-tuples with $k$ distinct entries, as "Mächtigkeit" explains. If so, what is the set of $k$-tuples in Definition 1.7 mean? Further, how to interpret ${\mathfrak G}^{[k]}$ as a group?

Answer 1

As you surmised, the geordnete Teilmengen der Mächtigkeit $k$ are the $k$-tuples of distinct elements of the set in question, here $\Omega$. In the definition of $k$-fold transitivity, $k$ is mentioned only in the subscripts of the tuples, but that's enough to make the definition say "A group $\mathfrak G$ of permutations of $\Omega$ is $k$-fold transitive if, for every two $k$-tuples $(a_1,\dots,a_k)$ and $(b_1,\dots,b_k)$ of distinct [within each $k$-tuple individually] elements of $\Omega$, there is an element $G\in\mathfrak G$ that sends each $a_i$ to the corresponding $b_i$." This agrees with the usual definition of $k$-fold transitivity.

Regardless of transitivity, any group $\mathfrak G$ that acts by permutations on $\Omega$ also acts on the set $\Omega^{[k]}$ of $k$-tuples of distinct elements of $\Omega$, namely by acting on all the components of the tuples. That group is called $\mathfrak G^{[k]}$. A more modern viewpoint would be that $\mathfrak G$ and $\mathfrak G^{[k]}$ are the same group with different actions (on different sets), but Huppert is thinking of the group as being a group of permutations, not as an abstract group equipped separately with an action.