# Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016 pp. 603–619.

Morava *K*-theory rings of the extensions of *C*_{2} by the products of cyclic 2-groups

**Authors**:
Malkhaz Bakuradze (1) and Natia Gachechiladze (1)

**Author institution:**(1) Iv. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences

**Summary: **

In 2011, Schuster proved that mod 2 Morava *K*-theory
*K*(s)^{*}(*BG*) is evenly generated for all groups *G* of order 32. There
exist 51 non-isomorphic groups of order 32. In a monograph by Hall
and Senior, these groups are numbered by 1, ..., 51. For the groups
*G*_{38}, ..., *G*_{41}, which fit in the title, the explicit ring structure is determined in a joint work of M. Jibladze and the author. In particular, *K*(s)^{*}(*BG*) is the quotient of a polynomial ring in 6 variables over
*K*(s)^{*}(pt) by an ideal generated by explicit polynomials. In this article
we present some calculations using the same arguments in combination
with a theorem by the author on good groups in the sense of
Hopkins–Kuhn–Ravenel. In particular, we consider the groups *G*_{36}, *G*_{37}, each
isomorphic to a semidirect product (*C*_{4}×*C*_{2}×*C*_{2}) ⋊ *C*_{2} , the group
*G*_{34} ≅ (*C*_{4}×*C*_{4}) ⋊ *C*_{2} and its non-split version *G*_{35}. For these groups the
action of *C*_{2} is diagonal, i.e., simpler than for the groups *G*_{38}, ..., *G*_{41},
however the rings *K*(s)^{*}(*BG*) have the same complexity.

2010 Math. Subj. Class. 55N20; 55R12; 55R40.

**Keywords:**Transfer, Morava

*K*-theory.

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