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August 23, 2019

Symmetric groups of order 30: TIFR GS 2018 Part A Problem 23

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]The permutation group \(S_{10}\) has an element of order 30.[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]TIFR 2018 Part A Problem 23[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Group Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Dummit and Foote[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

Start with hints

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Consider S={1,2,...,10}.\(S_{10}\) be the permutation group on S.
What will you do if one asked for a subgroup of order 3!=6?
  • We all would have taken the subgroup of \(S_3\) embedded in \(S_{10}\) right? Call the subgroup H taking identity transformation on {4,5,6,..10} and embedding in {1,2,3}.
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What do you do if one asked for a subgroup of order 5?
  • We will try to form a cyclic subgroup of order 5. We need to find a generator. Can you see it?
  • Keeping {1,2,.,5} same and taking 5 elements {6,7,..,10} and observe the permutation \(i\mapsto i+1\) for i=6,7,8,9 and \(10\mapsto 7\).Take the subgroup generated by this element.Observe this is a cyclic subgroup of order 5.Call this subgroup K.
  • Now any idea how to combine this?
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  • Observe that HK is a set of 30 elements. Does it seem HK is a subgroup?
  • Lemma: H and K are two subgroups of G. HK is a subgroup of G iff KH=HK.
  • Using the lemma prove that HK is really a subgroup of \(S_{10}\) of order 30.
  • Observe that the selection of disjoint elements of H and K is the main reason behind this!
  • Hence the answer is True.
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Watch the video

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