BFC-group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Symbol-free definition

A group is termed a BFC-group if there is a finite constant d such that no element in the group has more than d distinct conjugates. Such a group is also termed a d-BFC-group.

Any Abelian group is a 1-BFC-group.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Any subgroup of a BFC-group is a BFC-group. In fact, any subgroup of a d-BFC group is a d-BFC group. This follows from the fact that if two elements in the subgroup are conjugate in the subgroup, they are conjugate in the whole group. Hence, any conjugacy class in the subgroup is contained in a conjugacy class in the whole group.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
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A direct product of BFC-groups is a BFC-group. In fact, a direct product of a d_1-BFC-group and a d_2-BFC-group is a d_1d_2-BFC-group. This follows from the fact that if two elements in the direct product of G_1 and G_2 are conjugate, then their G_1-coordinates are conjugate and their G_2-coordinates are conjugate. Hence, conjugacy classes in G_1 \times G_2 are simply the pairwise direct products of conjugacy classes in G_1 and G_2.