GAP:IsAbnormal

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This article is about a GAP function.

This GAP function takes as input two groups, both having a common parent group. See other GAP functions like this.
This GAP function outputs a Boolean variable, i.e., it returns either true or false. View other GAP functions with Boolean output

Definition

Function type

This function takes as input a pair of two groups, both of which have a common parent group, and returns a Boolean variable (true/false).

Behavior

Suppose the first group is G and the second group is H.

  • If, for every g \in G, g is in the subgroup generated by H and the conjugate g^{-1}Hg = H^g, the function returns true. In particular, if H \le G, the function returns true if H is an abnormal subgroup of G.
  • If there exists g \in G such that g is not in the subgroup generated by H and H^g, the function returns false. In particular, if H is a subgroup of G that is not abnormal in G, the function returns false.

Typical use

IsAbnormal(group,subgroup);

Method

Code

IsAbnormal := function(G,H)
           return(ForAll(Set(G),g-> g in Group(Union(H,H^g))));
end;;