# Warm yourself with an MCQ

# What We Are Learning?

**Groups**are the main concept in abstract algebra here we will see about some application of

**subgroups**and

**cyclic groups**

# Understand the problem

(B) Every proper subgroup of \(\Bbb Z_n\) is cyclic

(C) Every proper subgroup of \(S_4\) is cyclic

(D) If every proper subgroup of a group is cyclic, then the group is cyclic.

##### Source of the problem

##### Topic

**Groups , Cyclic Group & Proper Subgroup**

##### Difficulty Level

##### Suggested Book

# Start with hints

So option (B) is correct. Now let prove that H \(\leq\) \(\mathbb{Z}_n\) = {\(\overline{0}\),\(\overline{1}\),…..,\(\overline{n-1}\)}. By well ordering principle H has a minimal non zero element ‘m’. Claim: H=<m> clearly <m> \(\subset\) H. For any r \(\in\) H by Euclid’s algorithm we have r=km+d where 0 \(\leq\) d < m which \(\Rightarrow\) d=r-km \(\in\) H If d \(\neq\) 0 then d<m which is a contradiction So, d=0 \(\Rightarrow\) r=km \(\Rightarrow\) H=<m> and we are done

# Knowledge Graph

# Some interesting Fact

# Some interesting Fact

Do you know that a cyclic group \(\Bbb Z_n\) can be seen inside a circle \(<e^{\frac{2\pi i}{n}}>\)? Below is one picture of \(\Bbb Z_8\) in the circle…

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

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