Reference request: automorphisms of a product of two cyclic groups of prime power order

[Arturo Magidin]

I'm looking for a reference to the following result (probably a
special case of a more general one):

Let p be a prime, and let a >= b > 0 be positive integers. let C_{p^a}
and and C_{p^b} be the cyclic groups of orders p^a and p^b,
respectively, and let A = C_{p^a} x C_{p^b} be their direct product,
with x generating the first factor and y generating the second. Then
every automorphism of A is of the form x |--> x^r y^s, y |-->
x^{mp^{a-b}} y^n, where r is determined modulo p^a, s, m, and n are
determined modulo p^b, and rn-smp^{a-b} is relatively prime to p.

A referee gave "Shoda (1928)" as a reference; I would appreciate both
a more modern one, and the full bibliographic details of that one.

Thank you in advance.

--
Arturo Magidin

[Derek Holt]

I guess this would be covered by the following references, which
handle split metacyclic p-groups. There is probably a better reference
for the abelian case!

MR2283679 (2007i:20055)
Bidwell, J. N. S.(NZ-OTG); Curran, M. J.(NZ-OTG)
The automorphism group of a split metacyclic $p$-group. (English
summary)
Arch. Math. (Basel) 87 (2006), no. 6, 488--497.
20E36 (20D45)

MR2322775 (2008d:20038)
Curran, M. J.(NZ-OTG)
The automorphism group of a split metacyclic 2-group. (English
summary)
Arch. Math. (Basel) 89 (2007), no. 1, 10--23.
20D45 (20D15)

MR2471983 (2010b:20050)
Bidwell, J. N. S.(NZ-OTG); Curran, M. J.(NZ-OTG)
Corrigendum to ``The automorphism group of a split metacyclic $p$-
group''. [Arch. Math. 87 (2006) 488--497] [ MR2283679]. (English
summary)
Arch. Math. (Basel) 92 (2009), no. 1, 14--18.
20E36 (20D45)

Derek Holt.