## Non-isomorphic 2-groups with isomorphic modular group algebras.(English)Zbl 1514.20019

Let $$k$$ be a field of characteristic $$p$$ and let $$G$$ and $$H$$ be finite $$p$$-groups. The modular isomorphism problem for group algebras, which was posed by R. Brauer in [Lect. Modern Math. 1, 133–175 (1963; Zbl 0124.26504)], asks whether an isomorphism between the group algebras $$kG$$ and $$kH$$ implies a group isomorphism between $$G$$ and $$H$$.
The main result of this impressive paper is the following.
Theorem. There are non-isomorphic finite 2-groups $$G$$ and $$H$$ such that the group rings of $$G$$ and $$H$$ over any field of characteristic 2 are isomorphic. In particular, the modular isomorphism problem has a negative answer.
The counterexamples constructed by the authors are so interesting that the reviewer cannot avoid reporting them. Let $$n$$ and $$m$$ be integers satisfying $$n > m > 2$$, then the groups given by the presentations: $G=\big \langle x,y,z \mid [x,y]=z,\; x^{2^{n}}=y^{2^{m}}=z^{4}=1, \; z^{x}=z^{-1}, \; z^{y}=z^{-1} \big \rangle$ $H=\big \langle a,b,c \mid c=[b,a],\; a^{2^{n}}=b^{2^{m}}=c^{4}=1, \; c^{a}=z^{-1}, \; c^{b}=c \big \rangle$ provide a negative answer to the modular isomorphism problem.
The smallest admissible values, $$n=4$$ and $$m=3$$, correspond to the groups identified in the library of small groups of GAP as $$[512,456]$$ and $$[512,453]$$. These groups have nilpotency class 3 and $$G'\simeq H'$$ are cyclic of order 4.
A consequence of the main theorem is the following Corollary: There are isomorphic 2-blocks of finite groups with non-isomorphic defect groups. In particular, the defect group of a 2-block is not determined by its Morita equivalence class over a finite field of characteristic 2.
The modular isomorphism problem remains open for various interesting classes of $$p$$-groups, including groups of nilpotency class 2 and groups of odd order.

### MathOverflow Questions:

Breakthroughs in mathematics in 2021

### MSC:

 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C20 Modular representations and characters

### Keywords:

modular isomoprhism problem

Zbl 0124.26504

### Software:

LAGUNA; SmallGrp; GAP
Full Text:

### References:

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