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LSpice
LSpice
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Finite groups with number of generators strictly lesserless than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $<a|a^n>$$\langle a \mid a^n\rangle$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is strictly bigger than the number of generators.

I want to get examples of $G$, where $G$ is a finite group with the property that for any presentation of $G$ the number of relations is strictly bigger than the number of generators.

More generally, can one put some condition on a finite group $G$, so that for any presentation the number of relations is strictly bigger than the number of generators?

Thanks in advance!

Finite groups with number of generators strictly lesser than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $<a|a^n>$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is strictly bigger than the number of generators.

I want to get examples of $G$, where $G$ is a finite group with the property that for any presentation of $G$ the number of relations is strictly bigger than the number of generators.

More generally, can one put some condition on a finite group $G$, so that for any presentation the number of relations is strictly bigger than the number of generators?

Thanks in advance!

Finite groups with number of generators strictly less than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $\langle a \mid a^n\rangle$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is strictly bigger than the number of generators.

I want to get examples of $G$, where $G$ is a finite group with the property that for any presentation of $G$ the number of relations is strictly bigger than the number of generators.

More generally, can one put some condition on a finite group $G$, so that for any presentation the number of relations is strictly bigger than the number of generators?

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gola vat
gola vat
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Finite groups with number of generators strictly lesser than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $<a|a^n>$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is strictly bigger than the number of generators.

I want to get examples of $G$, where $G$ is a finite group with the property that for any presentation of $G$ the number of relations is strictly bigger than the number of generators.

More generally, can one put some condition on a finite group $G$, so that for any presentation the number of relations is strictly bigger than the number of generators?

Thanks in advance!