Is it possible to use a sum function in a calculated column?
Yes, it's possible using a scalar valued function (scalar UDF) for you computed column but this would be a disaster. Using scalar UDFs for computed columns destroy performance. Adding a scalar UDF that accesses data (which would be required here) makes things even worse.
It sounds to me like you just need a good ol' fashioned index to speed things up. First some sample data:
IF OBJECT_ID('dbo.somelist','U') IS NOT NULL DROP TABLE dbo.somelist;
GO
CREATE TABLE dbo.somelist
(
ProductGroup INT NOT NULL,
[Month] TINYINT NOT NULL CHECK ([Month] <= 12),
Sales DECIMAL(10,2) NOT NULL
);
INSERT dbo.somelist
VALUES (1,1,22),(2,1,45),(2,1,25),(2,1,19),(1,2,100),(1,2,200),(2,2,50.55);
and the correct index:
CREATE NONCLUSTERED INDEX nc_somelist ON dbo.somelist(ProductGroup,[Month])
INCLUDE (Sales);
With this index in place this query would be extremely efficient:
SELECT s.ProductGroup, s.[Month], SUM(s.Sales)
FROM dbo.somelist AS s
GROUP BY s.ProductGroup, s.[Month];
If you needed to get a COUNT by month & product group you could create an indexed view like so:
CREATE VIEW dbo.vw_somelist WITH SCHEMABINDING AS
SELECT s.ProductGroup, s.[Month], TotalSales = COUNT_BIG(*)
FROM dbo.somelist AS s
GROUP BY s.ProductGroup, s.[Month];
GO
CREATE UNIQUE CLUSTERED INDEX uq_cl__vw_somelist ON dbo.vw_somelist(ProductGroup, [Month]);
Once that indexed view was in place your COUNTs would be pre-aggregated. You cannot, however, include SUM in an indexed view.