# Questions tagged [free-product]

In mathematics, specifically group theory, the free product is an operation that takes two groups $G$ and $H$ and constructs a new group $G\ast H$. The result contains both $G$ and $H$ as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties

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### The free product has the direct product as a factor group. What's the corresponding normal subgroup?

Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
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### Understanding quotient group in free product (van Kampen theorem)

I have never studied algebra thoroughly, so I do not have any experience with groups and no mindset that handles them naturally. For example, yesterday I have been looking into examples of quotient ...
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### Injectivity of maps in a free product group with amalgamation

Given countable groups $H, G_1$ and $G_2$ and group homomorphisms $\theta_i: H \rightarrow G_i$, we can construct the free group with amalgamation $(G_1 \ast G_2)/N$ where $N$ is the normal subgroup ...
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### Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

I’m trying to understand the differences between free products and direct products with an example: $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$. If I understand correctly, the ...
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### Ping-pong lemma assumptions

The ping-pong-lemma for subroups is often stated with the following assumptions, e.g. taking this one from Wikipedia: Let G be a group acting on a set $X$ and let $H_1, H_2, ..., H_k$ be subgroups of ...
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### If $A\ast_C B$ is finitely generated, are $A$ and $B$ finitely generated?

I know that if $A$ and $B$ are group of finite rank $n$, there is an amalgamated product $A\ast_{F_2} B$ of rank $2$. It is know that for every $n$ there exist groups $A$ and $B$ of rank $\ge n$ and ...
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