In Act 3, scene 1, what reasons does Kent give for the dissension between the Dukes and what implication does he leave in the mind of the Gentleman? 

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In this scene, Kent isn't so much discussing with the Gentleman the ways in which the Dukes of Cornwall and Albany are against each other, as the ways that it is suspected that they, together, are against the good of the land.  He does say that:

...There is division,

Although as yet the face of it be cover'd

Wtih mutual cunning, 'twixt Albany and Cornwall.

Yet, he goes on to spend much more time explaining how both houses are being watched by spies from France (where Cordelia is Queen).  It is trying to be discovered what might be at the bottom of the "snuffs and packings" (quarrels and intrigues) that have been discovered in both households.

Are they just picking on the old king, or do they have something "deeper" in mind?  The implication he leaves the Gentleman with is that this something "deeper" is treason against the country and that France will march against the Dukes to prevent it.

Kent will not reveal his identity to the Gentleman but gives him a ring to take to Dover to give to Cordelia, who will confirm that all that Kent has said is true.

So, though, Kent does not give reasons for the Dukes' dissension in this scene, he does warn the Gentleman that France is on the march to protect the realm against Albany and Cornwall, and that he (the Gentleman) must travel to Dover to meet Cordelia there in order to be of assistance to the King.