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$C [Q_{8}]$ and $C [Dih (C_{4})]$ both decompose as $C^{4} \times M_{2} (C)$

Example

Both order-eight nonabelian groups $Q_{8}$ and $Dih (C_{4})$ have the same complex Wedderburn type:

$C [Q_{8}] ≅ C^{4} \times M_{2} (C) and C [Dih (C_{4})] ≅ C^{4} \times M_{2} (C) .$

Facts & Assumptions

Given: The quaternion group $Q_{8} = {1, - 1, i, - i, j, - j, k, - k}$ and the dihedral group $D = Dih (C_{4}) = ⟨ r, s ⟩$ .

[L1]

The dihedral group $D$ has order $8$ , every element is $r^{a}$ or $r^{a} s$ with $0 \leq a < 4$ , and $r^{4} = s^{2} = 1$ with $s r s^{- 1} = r^{- 1}$ ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L2]

In $Q_{8}$ , the elements are ${1, - 1, i, - i, j, - j, k, - k}$ and the quaternion multiplication table gives $i^{2} = j^{2} = k^{2} = - 1$ , $i j = k$ , $j k = i$ , $k i = j$ , $j i = - k$ , $k j = - i$ , and $i k = - j$ (The quaternion group $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ inside the nonzero quaternions, The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ ).

[L3]

The group $Q_{8}$ has order $8$ , the element $- 1$ is its unique element of order $2$ , and each of $\pm i, \pm j, \pm k$ has order $4$ ( $Q_{8}$ is a subgroup of $H^{\times}$ with eight elements, and $- 1$ is its only element of order $2$ ).

[L4]

Over an algebraically closed field of characteristic prime to $∣ G ∣$ , the number of irreducible representations equals the number of conjugacy classes (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , the number of irreducible representations of $G$ equals the number of conjugacy classes).

[L5]

Under the same hypotheses, the irreducible degrees satisfy the sum-of-squares formula (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $\sum_{i} (\dim_{k} V_{i})^{2} = ∣ G ∣$ ).

[L6]

Under the same hypotheses, the group algebra is a product of full matrix algebras over the base field (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $k [G] ≅ \prod_{i = 1}^{r} M_{n_{i}} (k)$ ).

Verification

technique · direct

1.1

In $Q_{8}$ , the elements $1$ and $- 1$ are central by [L2]. Also $j i j^{- 1} = j i (- j) = (- k) (- j) = k j = - i$ , so $i$ is conjugate to $- i$ ; similarly $k i k^{- 1} = - i$ , and conjugation by $\pm 1$ or $\pm i$ fixes $i$ . Thus the conjugacy class of $i$ is ${i, - i}$ . The same calculation with cyclic permutations of $i, j, k$ gives the further classes ${j, - j}$ and ${k, - k}$ . Hence the conjugacy classes of $Q_{8}$ are ${1}, {- 1}, {i, - i}, {j, - j}, {k, - k} .$

L2L3givenalgebra

1.2

In $D$ , the relations of [L1] show that $r^{2}$ is central, while $s r s^{- 1} = r^{- 1} = r^{3}, r s r^{- 1} = r^{2} s, r (r s) r^{- 1} = r^{3} s .$ Therefore the conjugacy classes of $D$ are ${1}, {r^{2}}, {r, r^{3}}, {s, r^{2} s}, {r s, r^{3} s} .$ So both $Q_{8}$ and $D$ have five conjugacy classes.

L1givenalgebra

2.1

By [L4], each group has five irreducible complex representations. By [L5], their degrees $d_{1}, \dots, d_{5}$ satisfy $d_{1}^{2} + \dots + d_{5}^{2} = 8.$ Since every $d_{i} \geq 1$ , the only possible multiset is $1, 1, 1, 1, 2$ .

L4L5step 1.1step 1.2givenalgebra

3.1

Applying [L6] to each group gives one $2 \times 2$ factor and four $1 \times 1$ factors, so $C [Q_{8}] ≅ C^{4} \times M_{2} (C) and C [Dih (C_{4})] ≅ C^{4} \times M_{2} (C) .$

L6step 2.1givenalgebra∎

Depends on

Used by

False statement: a group with r conjugacy classes has an irreducible representation of degree r False statement

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof et al., Introduction to Representation Theory, Section 3.3 (standard reference, not scraped)
Peter Webb, A Course in Finite Group Representation Theory, Chapter 3 Section 3.4 (standard reference, not scraped)

C[Q8]C[Q8​] and C[Dih⁡(C4)]C[Dih(C4​)] both decompose as C4×M2(C)C4×M2​(C)