I am currently trying to understand the definition of the $\mathcal K$-residue and $\mathcal K$-radical of a group $G$. The definitions are from the book The Theory of Finite Groups by Kurzweil and Stellmacher.
Let $\mathcal K$ be always a class of groups that contains the trivial group and with a given group also all its isomorphic images. For any group $G$: $$ O^{\mathcal K}(G) := \bigcap_{A \unlhd G, \\ G/A \in \mathcal K} A \quad \mbox{and} \quad O_{\mathcal K}(G) := \prod_{A \unlhd G, \\ A \in \mathcal K} A $$ are characteristic subgroups of $G$. $O^{\mathcal K}(G)$ is the $\mathcal K$-residue and $O_{\mathcal K}(G)$ is the $\mathcal K$-radical.
They wrote they consider the following classes:
- $\mathcal A$ of all Abelian groups,
- $\mathcal N$ of all nilpotent groups,
- $\mathcal S$ of all solvable groups,
- $\mathcal P$ of all $p$-groups $(p \in \mathcal P)$,
- $\mathcal \Pi$ of all $\pi$-groups ($\pi \subset \mathcal P)$.
I am trying to grasp the meaning of these definitions, but what confuses me they give an example, they say: "If $G$ is a quaternion group of order $8$, then $O_{\mathcal A}(G) = G$".
So as I see it $O_{\mathcal K}(G)$ is a direct product over all abelian normal subgroups, in $Q_8$ they are $$\{ 1 \}, \{ 1, -1 \}, \{ \pm 1, \pm i \}, \{ \pm 1, \pm j \}, \{ \pm 1, \pm k \} $$ all are normal (for the ones with four elements because they have index is $2$ in $G$, the other by the "familiar" properties of $1$ and $-1$). But their direct product has $2 \cdot 4 \cdot 4 \cdot 4 > 8$ elements, so certainly this is not $Q_8$.
So I guessed maybe they meant just to take the product "up to isomorphism", meaning taking one isomorphic representive for each subgroup, in this case we have $$ \{ 1, -1 \} \times \{ 1, -1, i, -i \} $$ which has $8$ elements, but it is not isomorphic to $Q_8$, because \begin{align*} (1,1)(1,1) & = (1,1) \\ (1,-1)(1,-1) & = (1,1) \\ (-1,1)(-1,1) & = (1,1) \\ (-1,-1)(-1,-1) & = (1,1) \end{align*} and in $Q_8$ there are just two element which are self-inverse.
But I have no idea in what other way I should interpret the definition. Maybe someone can explain, give examples, or even give other characterisations of these sets (maybe in terms of some universal property) etc. Would be glad for each suggestion helping me grasping these concepts! Thank you!
Presumably by $\prod$ they mean an internal product of subgroups, so the radical is a subgroup.
If $N,M\triangleleft G$ are normal subgroups then $NM\triangleleft G$ is also a normal subgroup. Furthermore the product is independent of order, $NM=MN$. One easily extends this by induction to any finite number of normal subgroups, and their product is well-defined independent of specifying an order.