Group theory: compute, construct and prove structure
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 19 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
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Let G be a finite subgroup of SO(3) of order24 with no element of order6. Determine G up to geometric conjugacy and abstract isomorphism. Then determine the order of its full unit-quaternion preimage and prove that preimage is not G×C₂. Give an explicit lift that detects the difference.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let V be the twelve unit icosahedral vertices and H all orthogonal matrices preserving V. Prove H≅A₅×C₂ and |H|=120. Determine the kernel of its permutation action on the six unoriented vertex axes. Compare its faithful action on twelve vertices and explain why an orientation-reversing symmetry need not be a plane reflection.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For n≥3 let Dₙ≤O(2) be generated by rotation through 2π/n and S=diag(1,−1). Prove A↦diag(A,det A) embeds O(2) in SO(3), describe the embedded Dₙ and its pole-orbit data, and determine its full preimage under the unit-quaternion rotation map. Use r=cos(π/n)+k sin(π/n) and s=i. Can a lift of S have order two?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let V={(1,1,1),(1,−1,−1),(−1,1,−1),(−1,−1,1)}. Consider matrices obtained by an even permutation of coordinates followed by coordinate sign changes whose product is1. Prove these form a twelve-element rotation group preserving V and identify its faithful action on V. Compute its pole-orbit data, and find a unit-quaternion lift of the cyclic coordinate map (x,y,z)↦(z,x,y). Explain what this establishes and what it does not establish about all finite SO(3) subgroups.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Classify the groups C₅ ⋊ C₄ arising from all actions of C₄ on C₅. This is not a request to classify every group of order twenty. Write a normal-form law, identify which actions give isomorphic groups, and distinguish the resulting types using centers. Explain why arbitrary isomorphisms cannot identify any further types.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For each of C₈, C₄×C₂, D₄ and Q₈ choose the standard cyclic subgroup N of order four and analyze 1→N→G→C₂→1. Determine the conjugation action and whether the extension splits. Prove that the same kernel, quotient and conjugation action can occur in nonisomorphic groups. Explain why this does not contradict semidirect-product classification.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On the nonzero real numbers define a*b=ab/2. Prove this is a group and determine its identity and inverse formula. Prove the positive real numbers form a subgroup under this operation. Explain why importing the usual multiplicative identity without checking it would be wrong.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In the additive group Z/30Z find the order of 12 and every element of H=⟨12⟩. Describe all cosets without listing redundant representatives, and prove why each coset has the same number of elements. What does Lagrange alone say, and not say, about possible subgroups of a general group of order 30?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In S₅ let p=(1 3 5)(2 4) and q=(1 2 5), composing right to left. Compute pq in disjoint cycles, its inverse, order and sign. Justify the general rule for the sign of a k-cycle and explain why parity is well-defined despite different transposition decompositions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In D₅=⟨r,s:r⁵=e,s²=e,srs=r⁻¹⟩, calculate (r²s)(r⁴s). Prove the rotations form a normal subgroup and identify the quotient. Determine the center, checking both rotations and reflections.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In S₄ take H=⟨(1 2 3)⟩. Show why left-coset multiplication is not well-defined by conjugating with g=(1 4). Separately use the sign map to construct an isomorphism S₄/A₄≅C₂, proving representative independence and injectivity rather than comparing only sizes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In additive G=Z/36Z, let N=⟨12⟩ and K=⟨6⟩. Construct the map G/N→G/K and identify its kernel, proving (G/N)/(K/N)≅G/K. In a separate application, take H=⟨6⟩ and M=⟨9⟩ in G; compute H∩M, H+M and the two groups in the second isomorphism theorem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
List all abelian groups of order 72 up to isomorphism using primary factors. Compare A=C₈×C₃×C₃ and B=C₄×C₂×C₉ by maximum element order and number of elements killed by 3. Explain the completeness and uniqueness theorem behind the list.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let S₄ act on its six two-element subsets by permuting labels. Prove transitivity, identify the stabilizer of{1,2}, and determine whether the action is faithful. Separately find the conjugacy-class size and centralizer order of(1 2) in S₄, explaining the relevant orbit-stabilizer maps.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove every group of order 121 is abelian and classify the possibilities. Derive the nontrivial-center step from the class equation, and prove the needed claim that a cyclic quotient G/Z(G) forces G to be abelian.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let C₂ act on C³ by swapping the first two coordinates and fixing the third. Find the irreducible decomposition, construct invariant subspaces, and compute the dimension and block form of the equivariant endomorphism space. Verify it through the character inner product. Explain why scalar Schur cannot be applied to the whole three-dimensional representation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
S₃ acts on all 27 ordered triples of three letters, allowing repetition. Decompose this representation by its character and independently by classifying its orbits. Account for every orbit and show why the six-element orbits are regular.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let N be normal in G with N abelian and G/N≅S₄. Prove G is solvable without assuming N is central. Contrast this with any group surjecting onto A₅, proving why that case is impossible for a solvable group.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove every group of order 77 is cyclic. Then classify groups of order 10 up to isomorphism, explaining why exactly C₁₀ and D₅ occur rather than merely exhibiting these examples.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let G be a finite subgroup of SO(3) of order24 with no element of order6. Determine G up to geometric conjugacy and abstract isomorphism. Then determine the order of its full unit-quaternion preimage and prove that preimage is not G×C₂. Give an explicit lift that detects the difference.
Let V be the twelve unit icosahedral vertices and H all orthogonal matrices preserving V. Prove H≅A₅×C₂ and |H|=120. Determine the kernel of its permutation action on the six unoriented vertex axes. Compare its faithful action on twelve vertices and explain why an orientation-reversing symmetry need not be a plane reflection.
For n≥3 let Dₙ≤O(2) be generated by rotation through 2π/n and S=diag(1,−1). Prove A↦diag(A,det A) embeds O(2) in SO(3), describe the embedded Dₙ and its pole-orbit data, and determine its full preimage under the unit-quaternion rotation map. Use r=cos(π/n)+k sin(π/n) and s=i. Can a lift of S have order two?
Let V={(1,1,1),(1,−1,−1),(−1,1,−1),(−1,−1,1)}. Consider matrices obtained by an even permutation of coordinates followed by coordinate sign changes whose product is1. Prove these form a twelve-element rotation group preserving V and identify its faithful action on V. Compute its pole-orbit data, and find a unit-quaternion lift of the cyclic coordinate map (x,y,z)↦(z,x,y). Explain what this establishes and what it does not establish about all finite SO(3) subgroups.
Classify the groups C₅ ⋊ C₄ arising from all actions of C₄ on C₅. This is not a request to classify every group of order twenty. Write a normal-form law, identify which actions give isomorphic groups, and distinguish the resulting types using centers. Explain why arbitrary isomorphisms cannot identify any further types.
For each of C₈, C₄×C₂, D₄ and Q₈ choose the standard cyclic subgroup N of order four and analyze 1→N→G→C₂→1. Determine the conjugation action and whether the extension splits. Prove that the same kernel, quotient and conjugation action can occur in nonisomorphic groups. Explain why this does not contradict semidirect-product classification.
On the nonzero real numbers define a*b=ab/2. Prove this is a group and determine its identity and inverse formula. Prove the positive real numbers form a subgroup under this operation. Explain why importing the usual multiplicative identity without checking it would be wrong.
In the additive group Z/30Z find the order of 12 and every element of H=⟨12⟩. Describe all cosets without listing redundant representatives, and prove why each coset has the same number of elements. What does Lagrange alone say, and not say, about possible subgroups of a general group of order 30?
In S₅ let p=(1 3 5)(2 4) and q=(1 2 5), composing right to left. Compute pq in disjoint cycles, its inverse, order and sign. Justify the general rule for the sign of a k-cycle and explain why parity is well-defined despite different transposition decompositions.
In D₅=⟨r,s:r⁵=e,s²=e,srs=r⁻¹⟩, calculate (r²s)(r⁴s). Prove the rotations form a normal subgroup and identify the quotient. Determine the center, checking both rotations and reflections.
In S₄ take H=⟨(1 2 3)⟩. Show why left-coset multiplication is not well-defined by conjugating with g=(1 4). Separately use the sign map to construct an isomorphism S₄/A₄≅C₂, proving representative independence and injectivity rather than comparing only sizes.
In additive G=Z/36Z, let N=⟨12⟩ and K=⟨6⟩. Construct the map G/N→G/K and identify its kernel, proving (G/N)/(K/N)≅G/K. In a separate application, take H=⟨6⟩ and M=⟨9⟩ in G; compute H∩M, H+M and the two groups in the second isomorphism theorem.
List all abelian groups of order 72 up to isomorphism using primary factors. Compare A=C₈×C₃×C₃ and B=C₄×C₂×C₉ by maximum element order and number of elements killed by 3. Explain the completeness and uniqueness theorem behind the list.
Let S₄ act on its six two-element subsets by permuting labels. Prove transitivity, identify the stabilizer of{1,2}, and determine whether the action is faithful. Separately find the conjugacy-class size and centralizer order of(1 2) in S₄, explaining the relevant orbit-stabilizer maps.
Prove every group of order 121 is abelian and classify the possibilities. Derive the nontrivial-center step from the class equation, and prove the needed claim that a cyclic quotient G/Z(G) forces G to be abelian.
Let C₂ act on C³ by swapping the first two coordinates and fixing the third. Find the irreducible decomposition, construct invariant subspaces, and compute the dimension and block form of the equivariant endomorphism space. Verify it through the character inner product. Explain why scalar Schur cannot be applied to the whole three-dimensional representation.
S₃ acts on all 27 ordered triples of three letters, allowing repetition. Decompose this representation by its character and independently by classifying its orbits. Account for every orbit and show why the six-element orbits are regular.
Let N be normal in G with N abelian and G/N≅S₄. Prove G is solvable without assuming N is central. Contrast this with any group surjecting onto A₅, proving why that case is impossible for a solvable group.
Prove every group of order 77 is cyclic. Then classify groups of order 10 up to isomorphism, explaining why exactly C₁₀ and D₅ occur rather than merely exhibiting these examples.
