Union of proper subgroups

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If it is in H, then the element k must be in H, and vice versa. But this cannot be, as H and K are subgroups of GIn summary, the problem states that H and K are subgroups of G, and the task is to prove that their union is not equal to G. The hint provided is that a group cannot be the union of two proper subgroups. The proof involves considering elements in H and K that are not contained in the other subgroup.
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spotsymaj
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The following problem was given on a test of mine and I got it completely wrong. If anyone can help me with solving this problem that would be great

Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group cannot be the union of two proper subgroups
 
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spotsymaj said:
Let H and K be a subgroup of G, such that H is not equal to G and K is not equal to G . Prove that H union K is not equal to G. Hint: A group cannot be the union of two proper subgroups

If either H ≤ K or K ≤ H, then the desired result holds trivially. Now suppose that neither of these inclusions hold. Fix an element h in H not contained in K and another element k in K not contained in H. Is the product hk in either H or K?
 

1. What is the definition of a "Union of proper subgroups"?

The union of proper subgroups refers to the collection of all elements that are contained in at least one proper subgroup of a given group.

2. How is the union of proper subgroups different from the union of all subgroups?

The union of all subgroups includes all elements that are contained in any subgroup, including the entire group itself. The union of proper subgroups only includes elements that are contained in proper subgroups, which do not include the entire group.

3. Can the union of proper subgroups be a subgroup itself?

In general, the union of proper subgroups is not a subgroup of the original group. However, in certain cases where the group is finite, the union of proper subgroups may be a subgroup.

4. How can the union of proper subgroups be used in group theory?

The union of proper subgroups can be used to understand the structure of a group and its subgroups. It can also be used to prove properties of a group, such as showing that a group is not simple.

5. Is the union of proper subgroups commutative?

No, the union of proper subgroups is not commutative. The order in which the subgroups are combined can affect the elements that are included in the union.

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