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I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help.

Consider an object $A$ and a second object $B$(let them be groups if you so choose). We wish to consider and action of $A$ on $B$. Moreover there is a subobject $C \hookrightarrow B$(subgroup) which is annihilated by the action of $A$, i.e. the restriction of the action of $A$ on $B$ to $C$ sends $C$ to the zero object(the zero in $B$ which corresponds to the trivial group).

I thought it would be the kernel of the action, but this term is reserved for something else(in particular those objects which fix everything).

I think that this should be referred to as Torsion, and in particular, in the back of my mind, I keep thinking it is called the $A$-Torsion of $B$. But I am not sure.

Does anyone know what this has been called in the past?

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Both the group-theory and category-theory tags are not quite right, so I put both. :) – BBischof Oct 8 '10 at 21:00
What you wrote is a bit confused... What does it mean that «there is a subobject C↪B which is annihilated by the action of A, i.e. the restriction of the action of A on B to C sends C to the zero object»? How does the action map a subobject onto some other group? In the case of groups acting on groups (case of which I can make sense...) it is impossible for a subgroup to be "annihilated", as the group which acts acts by automorphisms. – Mariano Suárez-Alvarez Oct 8 '10 at 23:11
Maybe you can explain the concrete situation you have in mind, instead of trying to explain the abstract one? – Mariano Suárez-Alvarez Oct 8 '10 at 23:12
@Mariano Frankly, I don't really understand your first comment. But your second I do, so here is the point; given an R-bimodule M with the adjoint action, i.e. ad_r(m)=rm-mr for m an element of M, then we could ask for the sub-bimodule of M which is annihilated by this action, which in this case corresponds to those element of the module of which the right and left actions coincide. In this sense, there should be a subset of M(not necessarily a bimod itself) described by this condition. If I take the R-bispan of this subset I will get a sub-bimod. I hope this makes it clear. – BBischof Oct 9 '10 at 1:43
@Mariano Also, I phrased it in a slightly more general way, hoping that more users could contribute to the discussion. If I just talk about bimods or just talk about groups, I am being uselessly specific and I might exclude terminology from a setting which is slightly different. Sorry if it came across as hoity-toity or pretentious. – BBischof Oct 9 '10 at 1:46

in linear algebra, the subspace annihilated by a linear mapping $A$ is the nullity of $A$.

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The issue here is that A is a linear mapping acting on a space. In my situation I have two objects of the same category. In your case linear mappings are not in the category of vector spaces, however I appreciate your suggestion. – BBischof Oct 8 '10 at 21:23
Is "nullity" really used? Kernel seems to be universally used as far as I can tell. – Mariano Suárez-Alvarez Oct 8 '10 at 23:08
@Mariano: "Nullity" is used, but for the dimension of the kernel. "Nullspace", however, is very common (e.g., it occurs in Friedberg/Insel/Spence's book). – Arturo Magidin Oct 9 '10 at 2:31

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