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Cyclic Groups & Subgroups : IIT 2018 Problem 1

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Warm yourself with an MCQ

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What We Are Learning?

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Groups are the main concept in abstract algebra here we will see about some application of subgroups and cyclic groups
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Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" hover_enabled="0" box_shadow_style="preset2"]Which one of the following is TRUE? (A) $\Bbb Z_n$ is cyclic if and only if n is prime
(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.
[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.0.9" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.0.9"]IIT Jam 2018 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.0.9" open="off"]Groups , Cyclic Group & Proper Subgroup [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0.9" open="off"]EASY[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0.9" open="off"]ABSTRACT ALGEBRA BY DUMMIT AND FOOTE [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0.9"]We will solve this question by the method of elimination. Observe that if n is prime then $\mathbb{Z}_n$ is obviously cyclic as any of the subgroup <a> has order either 1 or n by Lagrange's theorem.Now if the order is 1 then a=id. So choose a($\neq$e) $\in \mathbb{Z}_n$ then |<a>|=n and <a> $\subseteq$ $\mathbb{Z}_n$ $\Rightarrow$ <a>= $\mathbb{Z}_n$. The problem will occur with the converse see $\mathbb{Z}_6$ is cyclic but 6 is not prime. In general $\mathbb{Z}_n$ = <$\overline{1}$> is always cyclic no matter what n is!! so option (A) is false. Can you rule out option (C)[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0.9"]Consider option (C) every proper subgroup of $S_4$ is cyclic. Consider { e , (12)(34) , (13)(24) , (14)(23) } = G  Observe that this is a subgroup and |G|=4. Moreover o(g)=2 $\forall$ g($\neq$e) $\in$ G So G is not cyclic. Hence option (C) is not correct. Can you rule out option (D)?

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0.9"]Consider $\mathbb{Z}_2$*$\mathbb{Z}_2$ which is also known as Klein's 4 group then it is not cyclic but all of it's proper subgroups are {0}*$\mathbb{Z}_2$ , $\mathbb{Z}_2$*{0} and {0}*{0} which are cyclic. Hence we can rule out option (D) as well.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0.9" content__hover_enabled="off|desktop"]

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

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Knowledge Graph

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Some interesting Fact

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Some interesting Fact

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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... [/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2020/01/Z-8.jpg" _builder_version="4.0.9"][/et_pb_image][et_pb_text _builder_version="4.0.9" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px||true|" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

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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.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/collegeprogram/" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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