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Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

EDIT: For both questions the answers below say YES. Here are two more complicated similar questions. Let $w$ be a group word with three variables $x,y,z$. Is the sentence $(\forall xy)(\exists z)w=1$ true in all groups if it is true in all finite groups? The same question for the sentence $(\exists x)(\forall yz)w=1$.

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

EDIT: For both questions the answers below say YES. Here are two more complicated similar questions. Let $w$ be a group word with three variables $x,y,z$. Is the sentence $(\forall xy)(\exists z)w=1$ true in all groups if it is true in all finite groups? The same question for the sentence $(\exists x)(\forall yz)w=1$.

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

I have reformulated the questions in more general form.
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owb
owb
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  • 7
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Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

EDIT: For both questions the answers below say YES. Here are two more complicated similar questions. Let $w$ be a group word with three variables $x,y,z$. Is the sentence $(\forall xy)(\exists z)w=1$ true in all groups if it is true in all finite groups? The same question for the sentence $(\exists x)(\forall yz)w=1$.

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.

EDIT: For both questions the answers below say YES. Here are two more complicated similar questions. Let $w$ be a group word with three variables $x,y,z$. Is the sentence $(\forall xy)(\exists z)w=1$ true in all groups if it is true in all finite groups? The same question for the sentence $(\exists x)(\forall yz)w=1$.

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owb
owb
  • 893
  • 7
  • 16

On sentences true in all finite groups

Let $w$ be a group word with two variables $x$ and $y$. Is the sentence $(\forall x)(\exists y)w=1$ true in every group if it is true in every finite group? The same question about the sentence $(\exists x)(\forall y)w=1$.