# Maximal and maximum subgroups of semigroups

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Can anyone tell me the difference between maximal and maximum subgroups of a semigroup, if there is one at all?

I seem to be a bit confused about if there is a difference at all. In my lecture notes I can only find references to maximal subgroups, whereas on a problem sheet I can only find references to maximum subgroups. I also vaguely remeber my lecturer emphasining a difference between maximum and maximal, but I was concentrating on the proof she'd just shown at the time, so I'm not sure if she was talking about a difference within semigroup theory or the difference between semigroup theroy and group theroy.

Any clarification would be great

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Doesn't anyone know?

I know semigroups are quite a specialist subject, but I thought someone migth know... ...please

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2 things spring to mind:

2. when you talk of maximal subgroups (of a group) they are not unique, not even up to order, it just means H<G and whenever H<=K<=G then K =H or K=G. Saying it is the maximum subgroup (apart from being grammatically dubious if we're thinking of maximum as being an adjective) has the air of it being unique. but i must stress this is just an opinion - there are few conventions universally and uniquely adopted in maths.

3. but you should ignore 2 and go with 1.

4. no really you should go with 1.

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I will go see my lecturer about it.

Only I though't I'd ask here first since it wasn't possible to go see the lecturer before the work had to be handed in (in the morning).

Anyway I'd already gone on with the assumption that maximum subgroup meant the largest and it seamed to work out fine for the questions I had. Though I'm still none-the-wiser about maximal v maximum since it turned out I didn't need any results directly relating to maximal subgroups for the questions. I guess I'll find out if I was right in a couple of days...whenever I get the work back or ask the lecturer...which ever is first.

Thanks again.

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• 1 month later...

There is a difference between maximal and maximum. A subgroup is maximal if there is no proper subgroup that contains it. I've rarely heard the term maximum applied to a subgroup, but what it should mean is that every other subgroup is contained in it. For example, in Z8, the subgroup {0, 2, 4, 6} is a "maximum subgroup." This is the same as the meaning of maximum applied to sets of real numbers: the maximum of the set [0, 3] is 3, because every other element of the set is less than it.

Note that a group might not have a maximum subgroup - for example, Z2 X Z2 - just as the set (0,3) does not have a maximum. (However, any group does have maximal subgroups.)

http://home.comcast.net/~paultrow/writing.htm

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