1MileCrash said:Homework Statement
Prove that G cannot have a subgroup H with |H| = n - 1, where n = |G| > 2.
Homework Equations
The Attempt at a Solution
Counter-example, the multiplicative group R and its subgroup, multiplicative group R+. Or, the additive group Z, and its subgroup of integer multiples of 2. What am I missing here? I think this is trivial for finite groups, but they don't say finite groups, they don't say anything.
A subgroup is a subset of a larger group that also forms a group under the same operation. It has the same properties as the larger group, but with a smaller set of elements.
To prove that a subset is a subgroup, you need to show that it satisfies three conditions: closure, associativity, and the existence of an identity element. This can be done by showing that the subset contains all possible combinations of elements from the larger group and that it obeys the same rules as the larger group.
Subgroup proof by verification is a method of proving that a subset is a subgroup by checking each of the three conditions (closure, associativity, and identity element) individually. This is done by checking every possible combination of elements from the subset and making sure that the result is also in the subset.
Yes, a subset can be a subgroup of multiple groups. This can happen if the subset satisfies the three conditions for multiple groups. In this case, the subgroup is called a common subgroup of those groups.
If a subset does not satisfy the three conditions for being a subgroup, then it is not a subgroup. In this case, the subset may still have some properties of the larger group, but it cannot be considered a subgroup.