. hho be means of examples that union of t o subgroups may or may not be subgroups
Question
Explain by means of examples that union of two subgroups may or may not be subgroups.
Solution
It seems like you're asking for examples to illustrate when the union of two subgroups is or is not a subgroup itself. Here are two examples:
- Union of Two Subgroups is NOT a Subgroup:
Consider two subgroups H and K of a group G. Let's say H = {e, a} and K = {e, b} where e is the identity, a and b are elements of G. The union of H and K is {e, a, b}. This set is not a subgroup of G because it's not closed under the operation of the group. For instance, if the operation is multiplication, then a*b is not in {e, a, b}.
- Union of Two Subgroups is a Subgroup:
This is only possible if one subgroup is contained within the other. For example, let H = {e, a} and K = {e, a, b} be subgroups of a group G. The union of H and K is K itself, {e, a, b}, which is a subgroup of G.
So, the union of two subgroups is a subgroup if and only if one subgroup is a subset of the other.
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