Consider the group G of non-zero real numbers under multiplication. Le...
The identity element of a group is an element such that combining it with any other element in the group leaves the other element unchanged. The inverse element is the element that, when combined with another element, produces the identity element. Therefore, the identity element of a group has an inverse element.
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Consider the group G of non-zero real numbers under multiplication. Le...
√2 ε H unit

Consider the group G of non-zero real numbers under multiplication. Le...
To show that H is a subgroup of G, we need to show that it satisfies the three conditions for being a subgroup:
1. H is non-empty: Since H contains the element 1, which is a non-zero real number, it is non-empty.
2. H is closed under multiplication: Let x and y be any two elements in H. We need to show that their product xy is also in H. Since x and y are non-zero real numbers, their product xy is also a non-zero real number. Therefore, xy is in H.
3. H is closed under taking inverses: Let x be any element in H. We need to show that its inverse 1/x is also in H. Since x is a non-zero real number, 1/x is also a non-zero real number. Therefore, 1/x is in H.
Since H satisfies all three conditions, it is a subgroup of G.