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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. :)2010-10-08
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    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.2010-10-08
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    Maybe you can explain the concrete situation you have in mind, instead of trying to explain the abstract one?2010-10-08
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    @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.2010-10-09
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    @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.2010-10-09
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    @BBischof, the thing is, what you generalized into is difficult to make sense of (in a general category, there is no sense in which an object can act on another, and "annihilation" does not make sense in a general category, either; &c.) . If you want a more general question, I think it would be optimal to state in detail the situation you *do* know of, and ask for what is a more general situation extending yours.2010-10-09

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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.2010-10-08
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    Is "nullity" really used? *Kernel* seems to be universally used as far as I can tell.2010-10-08
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    @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).2010-10-09