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xiavatar
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Suppose θ: A → B is a homomorphism. And assume S ≤ B. Is it necesarily true that if S is a subgroup, that is not completely contained in the range, its preimage forms a subgroup?
xiavatar said:I used subgroup criterion test, and it should be a subgroup. But i just wanted to make sure I didn't miss anything trivial.
The preimage of an arbitrary subgroup is a subset of the domain of a function that maps to all the elements in the subgroup. In other words, it is the set of all elements in the domain that, when applied to the function, will produce an element in the subgroup.
To calculate the preimage of an arbitrary subgroup, you need to determine all the elements in the domain that, when applied to the function, will produce an element in the subgroup. This can be done by finding the inverse of the function and then applying the elements of the subgroup to the inverse function.
The preimage of an arbitrary subgroup is important in abstract algebra and group theory as it helps to understand the structure of a group and its subgroups. It also allows for the identification of elements in the domain that map to specific elements in the subgroup.
Yes, it is possible for the preimage of an arbitrary subgroup to be empty. This means that there are no elements in the domain that, when applied to the function, will produce an element in the subgroup. This can happen if the subgroup is not a valid subset of the codomain of the function.
The concept of preimage of an arbitrary subgroup is used in various fields such as cryptography, data compression, and coding theory. It helps in finding efficient ways to encode and decode information by identifying patterns and structures within a group and its subgroups.