Practice: Groups · The Lumenary

Groups

Warm-upHuman

What is the order of the permutation $\sigma = (1\,2\,3)(4\,5)$ in $S_5$ ?

Groups

Warm-upHuman

What is the order of the element $8$ in the additive group $\Z/12\Z$ ?

Groups

CoreHuman

A group $G$ has order $20$ . Which of the following cannot be the order of a subgroup of $G$ ? (Select all that are impossible.)

Choose all that apply

Groups

CoreHuman

A student writes: “ $|S_3| = 6$ and $3 \mid 6$ , so by Lagrange's theorem $S_3$ has an element of order $3$ .” What is the best assessment?

Choose one

The argument is correct.The conclusion is true, but Lagrange's theorem does not give it: Lagrange only says which orders are possible. Cauchy's theorem (for primes dividing

|G|

) or the explicit 3-cycle

(1\,2\,3)

justifies it.The conclusion is false:

S_3

has no element of order

3

.Lagrange's theorem applies only to abelian groups, and

S_3

is not abelian.

Groups

CoreHuman

Which of these subgroups of $S_3$ are normal? (Select all that apply.)

Choose all that apply

\{e\}

A_3 = \{e, (1\,2\,3), (1\,3\,2)\}

\{e, (1\,2)\}

S_3

Groups

CoreHuman

Use the orbit–stabiliser theorem: how many rotational symmetries does a regular dodecahedron (12 regular pentagonal faces) have?

Groups

StretchHuman

Each face of a cube is painted black or white. How many colourings are different up to rotation of the cube? (Use Burnside's lemma with the $24$ rotations.)

Groups

CoreFormal

Which Lean statement correctly formalises “a subgroup of index $2$ contains the square of every element of the group”? (G is a group and H : Subgroup G.)

Choose one

theorem t (h : H.index = 2) (g : G) : g ^ 2 ∈ Htheorem t (h : H.index = 2) (g : G) (hg : g ∈ H) : g ^ 2 ∈ Htheorem t (h : Nat.card H = 2) (g : G) : g ^ 2 ∈ Htheorem t (g : G) (h : (g ^ 2) ∈ H) : H.index = 2

Groups

StretchComputational

A permutation of $\{0, 1, \dots, n-1\}$ is given as a tuple p with p[i] the image of i. Write perm_order(p) returning its order, by finding the lengths of its disjoint cycles and taking their least common multiple. (You may use math.lcm.)

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