|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
Is a trivial subgroup a proper subgroup? The article says so but my text says otherwise. -- Taku 23:07, 8 October 2005 (UTC)
- I think it is the same as with a set. A proper subset Y of X is a subset Y which does not equal X. So, unless the group is trivial to start with, its trivial subgroup must be proper. No? Oleg Alexandrov (talk) 00:09, 9 October 2005 (UTC)
- I think I have seen both, but what Oleg writes is what I would guess is the most common. It needs a survey of recent literature. People write "non-trivial proper subgroup" when they want to make sure there is no misunderstanding. --Zero 11:07, 10 October 2005 (UTC)
No, it is. The element 1 generates the group.
subgroup vs. uninion
- The union of two groups isn't a group (unless one of the two is a subgroup of the other). The direct product may be what you're looking for. --Zundark (talk) 09:24, 8 April 2011 (UTC)
subgroup operation or product
If in the definition of group one speaks of a generic operation *, why is so that in the definition of subgroup (or rather in the explanations below the definition, at the first property of subgroups) this operation has become a product? Is it language misuse, or is it a requirement?
Put another way: If the set of real numbers R forms a group under the operation +, with identity 0 and inverse -a, is the set of integers Z a subgroup of R under the operation +?
If so: can we speak of operator + as a product? This would be very confusing, although I could admit it as a convenient shortcut. However, in Wikipedia, we may want to guide the unexpert reader a little better.