Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Statement

Suppose A and B are (possibly equal) nontrivial groups. Then, there exist groups such that all the following conditions are satisfied:

• H is a normal subgroup of K and the quotient group K / H is isomorphic to A.
• K is a normal subgroup of G and the quotient group is isomorphic to B.
• H is not a normal subgroup of G.

Related facts

• Normality is not transitive
• Conjunction of normality with any nontrivial finite-direct product-closed property of groups is not transitive
• Normality is not transitive for any nontrivially satisfied extension-closed group property
• There exist subgroups of arbitrarily large subnormal depth

Proof

The construction is as follows. Let G be the wreath product of A and B for the regular group action of B. Let K be the subgroup A^B, i.e., the normal subgroup that forms the base of the semidirect product, and let H be the subgroup of K where a particular coordinate is the identity element. (If we are thinking of A^B as functions from B to A, then H can be taken as the subgroup comprising those functions that send the identity element of B to the identity element of A -- here, the particular coordinate becomes the coordinate corresponding to the identity element of B).

Thus, K is isomorphic to . Then:

• H is normal in K and K / H is isomorphic to A: Fill this in later
• K is normal in G and G / K is isomorphic to B: Fill this in later
• H is not normal in G: The action of B on the coordinates in K = A^B is transitive on the coordinates, so H is not preserved under this action.