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The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to 
$\sim\!5$
yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of 
$\sim\!162$
h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of 
$0.24\ \mathrm{mJy\ beam}^{-1}$
and angular resolution of 
$12-20$
arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. The total area of the pilot survey footprint is 5 131 square degrees, covering six distinct regions of the sky. An initial search of two of these regions, totalling 1 646 square degrees, revealed 28 highly variable and/or transient sources. Seven of these are known pulsars, including the millisecond pulsar J2039–5617. Another seven are stars, four of which have no previously reported radio detection (SCR J0533–4257, LEHPM 2-783, UCAC3 89–412162 and 2MASS J22414436–6119311). Of the remaining 14 sources, two are active galactic nuclei, six are associated with galaxies and the other six have no multi-wavelength counterparts and are yet to be identified.
Morgan and Parker proved that if G is a group with 
${\textbf{Z}(G)} = 1$
, then the connected components of the commuting graph of G have diameter at most 
$10$
. Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a 
$2$
-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most 
$8$
. We prove that the hypothesis 
$Z (G) = 1$
in these results can be replaced with 
$G' \cap {\textbf{Z}(G)} = 1$
. We also prove that if G is solvable and 
$G/{\textbf{Z}(G)}$
is either a Frobenius group or a 
$2$
-Frobenius group, then the commuting graph of G is disconnected.
