Algorithm to find the size of a quotient of a free group

Are there any algorithms to find the size of an algebraic quotient of a free group? It would take the generators as input and output the size. For example, an input could be something like {a,b: a^8=b^2=1, ab=ba^3} (a quasidihedral group), and the output would be the integer 16, since there are 16 different ways to write a string formed of a’s and b’s if you treat aaaaaaaa and bb as the empty string, and the string ab as the same as the string baaa.

Ideally the algorithm would also be able to say if a group was infinite. It’s not obvious by looking at the presentation what the size of the group is, and I think it’s a fairly interesting question that I’ve comes across a few times in my algebra class, so I was surprised that I couldn’t find a tool to solve it for me or any papers talking about it, it seems that computers should be good at this. Wondering if anyone has any good ideas.

How can I create the context free grammar for this language? [duplicate]

I need help finding the context-free grammar for this language.

$ $ L = \{a^ib^jc^k \in \{a,b,c\}^* \mid \text{$ i,j,k \geq 1$ , and $ i=j$ or $ i=k$ or both}\}. $ $

I’ve found a way to satisfy $ i = j$ or $ i = k$ or both, and $ i,j,k \ge 0$ .

But I am lost at the $ i,j,k \ge 1$ part. Any tips or help would be greatly appreciated.

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As a DM how can i manage a group with different amount of free time?

I’ve started playing D&D 5e with a group of friend recently. I’m the DM and there are 4 players. In this time of crisis, one is still working and is only available during week ends and the other 3 (and me) have quite a lot of spare time and are eager to play on a daily basis.

Is there a viable option for me and the 3 players to do some side questing or something else and play while not creating a gap between the players ?

Solutions like doing a side campaign without earning neither experience point nor gold seems to ruin the experience. I thought of multi classing but it seems to really impact the character even if the player only use its original class.

I see 3 solutions so far, but none are satisfying : 1: make player create another character 2: create a copy of the current character which will evolve separately 3: do an adventure where all exp and gold won’t be carried over to the main adventure

So i wondered if there is anything that i didn’t’ think of. Sorry if this is a duplicate but i didn’t find anything cause i don’t really know how to write the question in the first place.

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