Geodesic
The Cayley graph of a~subgroup of the Burnside group~$B_0(2,5)$
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 19-21

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Let $B_0(2,5)=\langle a_1,a_2\rangle$ be the largest two-generator Burnside group of exponent five. It has the order $5^{34}$. We define an automorphism $\varphi $ under which every generator is mapped into another generator. Let $C_{B_0(2,5)}(\varphi)$ be the centralizer of $\varphi$ in $B_0(2,5)$. It is known that $|C_{B_0(2,5)}(\varphi)|=5^{17}$. We have calculated the growth function of this group relative to the minimal generating set $X$. As a result, the diameter and the average diameter of $C_{B_0(2,5)}(\varphi)$ are computed: $D_X(C)=33$, $\overline D_X(C)\approx26{,}1$.
Keywords: Burnside group, Cayley graph, growth function. 
@article{PDMA_2017_10_a5,
     author = {A. A. Kuznetsov and A. S. Kuznetsova},
     title = {The {Cayley} graph of a~subgroup of the {Burnside} group~$B_0(2,5)$},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {19--21},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a5/}
}
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A. A. Kuznetsov; A. S. Kuznetsova. The Cayley graph of a~subgroup of the Burnside group~$B_0(2,5)$. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 19-21. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a5/