Agitator does not turn ..... you can hold on to & it doesn't turn in wash cycle
This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.
First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action
Enter your model number (110.#######) and you'll see a list of major sub-components with diagrams and parts.
If you move the timer to the spin cycle and the washer drains and spins, the motor coupler (A common replacement part) is OK.
This points to one of two possible solutions.
First, I'd remove the agitator(s) and look inside at the splines inside the agitators and at the splines on the main agitator shaft.
Here is a description of how to remove the agitators. Your washer may be a bit different but the Sears site should help to determine how to do it.
http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=136&osCsid=7a980a8b05b84ff3b7ee1c8ba46e2ae8
If yoy have a two piece agitator and only the top section doesn't agitate, the problem is likely the agitator "dogs" as described in the illustrations.
If the Agitator splines are worn or damaged, You'll have to replace the agitator.
If the splines on the transmission agitator shaft are worn, you'll probably need a new or rebuilt transmission.
In addition, see the following Whirlpool / Kenmore service manual and parts list for their commercial washers (consumer models are basically the same).
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf
http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%20W10114743.pdf
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