Solvable group

General groups

Definition. A group is solvable if we find a finite sequence of normal subgroups $(A_0, \ldots, A_r)$ such that

$$ I=A_{0} \triangleleft A_{1} \triangleleft \ldots \triangleleft A_{r}=G $$

($\triangleleft$ means "is normal subgroup of") and such that

$$ A_{k+1}/A_{k} $$

is abelian.

$\blacksquare$

Remarks:

The sequence $(A_0, \ldots, A_r)$ is called normal series. If each term of the series is normal not in the whole group but only in the preceding term, then the series is called subnormal series.

It turns out that it is enough the existence of a subnormal series with abelian normal factors (see here

In the case of Lie groups we have this slightly different definition:

Definition (@olver86). Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Then $G$ is solvable if there exists a chain of Lie subgroups

$$ \{e\} = G_0 \subseteq G_1 \subseteq G_2 \subseteq \ldots \subseteq G_{r-1} \subseteq G_r \subseteq G $$

such that for each $k = 1, \ldots, r$, $G_{k-1}$ is a $k$-dimensional subgroup of $G$ and $G_{k-1}$ is a normal subgroup of $G_k$.

Equivalently, $\mathfrak{g}$ is a solvable Lie algebra.

$\blacksquare$

Taking into account the normal subgroup vs ideal of Lie algebra relationship, it looks intuitive understand the equivalences in the definition above.

On the other hand, a solvable Lie group is solvable like a general group since (I think) every 1-dimensional Lie group is abelian.

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Author of the notes: Antonio J. Pan-Collantes

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