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Hi.

What's the difference between together and eachother?

I can use them without any problems, but when I come to explain my students, my mind goes blank.

Your help and time are highly appreciated in advance

Regards
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'Each other' refers to a reciprocal action: I do X to you and you do X to me; we do X to each other.

'Together' refers to a common action: You and I do X to someone or something else; we do X together.
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Thanks. That was a good explanation.

We can always replace together with with eachother. Am I right?

Ex. We go swimming together.

Ex. We go swimming with each other.

Thanks
We can always replace together with with each other. Am I right? -- That sounds reasonable, but this being English 'always' is a dangerous word.

We danced together / with each other. -- Fine!

We went bungee-jumping together /... with each other?
Thanks Mister Micawber.

I agree with you completley. Always and never are dangerous words, just one counter example can prove one wrong.
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You Can say stop hitting each other but you Can not say stop hitting together . I guess you use together when you do something together with somebody else & each other when you do something to each other , where you alone are doing something but that other person do it back to you