# 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.

## 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 $G_2$.