View source for Tour:Union of two subgroups is not a subgroup unless they are comparable

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{{guided tour|beginners|Introduction three|Left coset of a subgroup|Intersection of subgroups is subgroup}}
{{quotation|We just saw that an intersection of subgroups is always a subgroup. The corresponding statement isn't true for unions.<br>'''WHAT YOU NEED TO DO''':
* Read, and try to prove, the statements below
* Read the proof after you've tried
* Try to come up with an example (think of the group of integers, for instance) to illustrate this.}}
{{#lst:Union of two subgroups is not a subgroup unless they are comparable|beginner}}
{{quotation|'''PONDER (WILL BE EXPLORED LATER IN THE TOUR)''':
* What are the key facts about the structure of groups that we're using here?
* For what other weaker structures (magma, monoid, semigroup, quasigroup) does the argument go through?<br>
'''WHAT'S MORE''': Below are listed some related facts, counterexamples for more than two subgroups, and the relation with a result in commutative algebra. Ignore the parts that use unfamiliar terminology.}}
{{#lst:Union of two subgroups is not a subgroup unless they are comparable|revisit}}
{{guided tour-bottom|beginners|Introduction three|Left coset of a subgroup|Intersection of subgroups is subgroup}}