# Meaning of colon in group theory, if not subgroup index?

• AgentBased
In summary, the paper discusses the use of colons in subgroup descriptions, specifically in the form of (G_1 x G_2):G_3. The use of a colon in this context does not refer to the index of a subgroup, as it typically does, but rather indicates a semi-direct product. This notation is derived from the computational algebra system GAP.
AgentBased
I am reading a paper where the author uses colons in the description of groups. Example (not verbatim): "This subgroup is isomorphic to (Z_5 X A_4):Z_2". Several subgroups are described in the same way (as (G_1 x G_2):G_3) throughout the paper.

I have seen the colon in G to indicate the index of a subgroup H, in G, but that doesn't seem to make sense in this context. Does anyone know what this means?

Perhaps /, as in quotient group G/H? (they both suggest some sort of division)

Evidently, it turns out that it's semi-direct product. The notation comes from the computational algebra system GAP.

## 1. What is the meaning of a colon in group theory?

A colon in group theory represents the notation for a subgroup index, which is the number of cosets in a subgroup that make up the original group. It is used to show the relationship between a subgroup and the original group.

## 2. How is a colon used in group theory?

In group theory, a colon is used to indicate the subgroup index, which is written as [G:H]. This notation represents the number of cosets in the subgroup H that make up the group G.

## 3. What does the subgroup index represent?

The subgroup index, written as [G:H], represents the number of cosets in the subgroup H that make up the group G. It is used to understand the structure and size of a group and its subgroups.

## 4. Can the subgroup index be any number?

No, the subgroup index must be a positive integer. This makes sense because the number of cosets in a subgroup cannot be negative or a fraction. Additionally, the subgroup index cannot be greater than the order of the group.

## 5. How is the subgroup index related to the order of the group and subgroup?

The subgroup index is related to the order of the group and subgroup in the following way: [G:H] = |G|/|H|. This means that the subgroup index is equal to the order of the group divided by the order of the subgroup.

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