# Bounds for Khukhro-Makarenko theorems

Let’s define the set of outer-commutator group words $$OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$$ using the following recurrence:

$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$

$$\forall u, v \in OC \text{ } [u, v] \in OC$$

Let’s call a group variety outer commutator variety iff it can be defined by a single outer commutator identity. Examples of outer commutator varieties include the varieties of

a) all $$n$$-step nilpotent groups (defined by $$[…[[x_0, x_1], x_2]… x_n]$$)

b) all $$n$$-Engel groups (defined by $$[…[[x_0, x_1], x_1]… x_1]$$)

c) all $$n$$-step soluble groups (defined by $$[…[[x_0, x_1], [x_2, x_3]]…, ...[[x_{n-1}, x{n-2}],[x_{n-1}, x_n]]…]$$

d) the trivial group (defined by $$x_1$$)

e) all groups (defined by $$[x_1, x_1]$$)

Also, the class of outer commutator varieties is closed under variety product.

There are three theorems proved by Eugeny Khukhro and Natalya Makarenko about outer commutator varieties:

First Khukhro-Makarenko Theorem

Suppose $$\mathfrak{U}$$ is an outer commutator variety, $$G$$ is a group and exists such subgroup $$H \leq G$$, that $$H \in \mathfrak{U}$$ and $$[G:H] < \infty$$. Then exists a subgroup $$N \leq G$$, such that $$N$$ is characteristic $$G$$, $$[G:N] < \infty$$ and $$N \in \mathfrak{U}$$.

Second Khukhro-Makarenko Theorem

Suppose $$\mathfrak{U}$$ is an outer commutator variety, $$G$$ is a group and exists such subgroup $$H \leq G$$, that $$|V_\mathfrak{U}(H)| < \infty$$ and $$[G:H] < \infty$$. Then exists a subgroup $$N \leq G$$, such that $$N$$ is characteristic $$G$$, $$[G:N] < \infty$$ and $$|V_\mathfrak{U}(H)| < \infty$$.

Third Khukhro-Makarenko Theorem

Suppose $$\mathfrak{U}$$ is an outer commutator variety, $$G$$ is a group and exists such subgroup $$H \leq G$$, that $$V_\mathfrak{U}(H)$$ is locally finite and $$[G:H] < \infty$$. Then exists a subgroup $$N \leq G$$, such that $$N$$ is characteristic $$G$$, $$[G:N] < \infty$$ and $$V_\mathfrak{U}(H)$$ is locally finite.

Here $$V_\mathfrak{U}$$ stands for the corresponding verbal subgroup.

For the First Khukhro-Makarenko theorem, there is an upper bound proved by Anton Klyachko and Yulia Melnikova:

If $$\mathfrak{U}$$ is defined by an outer commutator group word $$w$$, then

$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!)$$

where

$$f(x) = x(x+1)$$

$$f^{(n)}(x) = \begin{cases} x & \quad n = 0 \\ f(f^{(n-1)}(x)) & \quad n > 0 \end{cases}$$

$$d(x_i) = 1$$

$$d([v, u]) = d(v) + d(u)$$

For the conditions of the second Khukhro-Makarenko theorem a similar bound can be derived:

$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!) + ln(|V_\mathfrak{U}(H)|)$$

My question is:

Are there any similar upper bounds for $$[G: N]$$ under conditions of the Third Khukhro-Makarenko theorem?

Personally, I failed to find anything for that third case.

The Second, Third, and many others "Khukhro--Makarenko theorems" were proved here, see also arXiv (so, they are rather "Klyachko--Milentyeva theorems"). The bound is always the same: $$\log_2|G:N|\leqslant f^{d(w)-1}(\log_2(|G:H|!)).$$ (In your question, there are several misprints in the estimates.) These facts are special cases of The Large-Subgroup Theorem from the paper cited above.