# 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…

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#### College Mathematics Program

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