Semidirect product notation convention

Question

I was taught that if $G \simeq H \ltimes K$, then (by convention) $H$ is the subgroup that is normal, but I see on Wikipedia and elsewhere that other people use the convention that the above indicates that $K$ is the one that is normal.

Has anyone encountered any literature where $H$ is the one that is normal, or do most people follow the convention that $K$ is normal?

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Edit for clarification

$G \simeq H \ltimes K \simeq K \rtimes H $

• Convention 1: $H$ is normal
• Convention 2: $K$ is normal

Answer 1

I've always connected it with the symbol for normal group: $\unlhd$. I assumed that in both cases the arrow points towards the normal subgroup. I've seen a reversed version of your symbol with the normal subgroup on the left, but the arrow always pointed the right way.

Answer 2

Note that both conventions make sense. It seems that there is no disagreement about the fact that the triangle in the notation is connected to the notation for normal subgroup: $H \trianglelefteq G$ or $H \vartriangleleft G$.

With convention 1 the triangle is closer to the normal subgroup, but the tip is pointing the other direction, while with convention 2, the tip is pointing the right way, but the triangle is closer to the other subgroup.

I've got the impression (but it is just impression) that convention 2 is used more often. On the other hand, for example the names of the corresponding Unicode characters (U+22C9, U+22CA) use convention 1.

Note that there is also an ortogonal question in the used conventions – wherether the normal subgroup is the left or the right factor in the product. Here it seems that it is usually the left one.