
What is the number of improper subgroups in a group?
A. \[2\]
B. \[3\]
C. depends on the group
D. \[1\]
Answer
161.4k+ views
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.
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.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2026 Syllabus PDF - Download Paper 1 and 2 Syllabus by NTA

JEE Main Eligibility Criteria 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
