Question

What is the number of improper subgroups in a group?
A. \[2\]
B. \[3\]
C. depends on the group
D. \[1\]

Answer 1

Hint: A subset \[H\] of a group \[G\] is a subgroup of \[G\] if \[H\] is itself a group under the same binary operation in \[G\]. An improper subgroup of a group means a subgroup which is not equal to the group. An improper subgroup contains fewer elements than that of a group.

Complete step-by-step solution:
Every group has an identity element.
Let \[e\] be the identity element in the group \[G\].
Since, identity element is unique in a group, so every subgroup of a group must contain the same as its identity element.
So, the subgroup \[\left\{ e \right\}\], containing only the identity element, is an improper subgroup of a group.
Every group is a subgroup of itself and it is improper.
All other subgroups other than the group formed by the identity element and the group itself are proper subgroups.
So, the number of improper subgroups in any group is \[2\].
Hence option A is correct.

Additional Information:
Group: A group is a set that has an operation that allows any two elements to be connected to form a third element in such a way that the operation is associative, an identity element will be defined, and each element has its inverse.
Subgrup: A subgroup is a part of a larger group, which is a group. In other words, if H is a group and G is a non-empty subset of H, then H is referred to as G's subgroup.

Note: The number of improper subgroups in any group is \[2\] but the number of proper subgroups in a group depends on the group.